Econ 320B, Set 1 - University of Birminghamsocscistaff.bham.ac.uk/jensen/320b-Set-1-beamer.pdf · Outline Practical Information Course Overview Course Overview Method of Assessment

OutlinePractical Information

Course OverviewCourse Overview

Method of AssessmentExchange Economies, Basic Notation

Perfect Competition

Econ 320B, Set 1

Martin K. Jensen (U. B’ham)

Martin K. Jensen (U. B’ham) Econ 320B, Set 1




Perfect Competition

1 Practical Information

2 Course Overview

3 Course Overview

4 Method of Assessment

5 Exchange Economies, Basic Notation

6 Perfect Competition





Perfect Competition

Lecturer: Dr. Martin Kaae Jensen, Economics Department(JG Smith Building) Room 206.

WebCT is not used. All information about the course can befound on the course homepage which is located at:http://socscistaff.bham.ac.uk/jensen/ECON320B.htm

Teaching material:

Jehle and Reny, Advanced Microeconomic Theory, chapter 5.Lecture notes on social choice theory.Occasional lecture notes.


http://socscistaff.bham.ac.uk/jensen/ECON320B.htm




Perfect Competition



Teaching material:







Perfect Competition



Teaching material:







Perfect Competition

Course overview:

1 General equilibrium in exchange economies. Properties ofexcess demand functions. Existence of equilibrium. Efficiency.

2 General Equilibrium with production in private ownershipeconomies and under lump-sum transfers.

3 The welfare theorems. The core of a competitive economy.4 Limitations of classical welfare analysis.5 Social choice theory.

There’s quite a bit of mathematics in this material...





Perfect Competition

Course overview:

1 General equilibrium in exchange economies. Properties ofexcess demand functions. Existence of equilibrium. Efficiency.

2 General Equilibrium with production in private ownershipeconomies and under lump-sum transfers.

3 The welfare theorems. The core of a competitive economy.4 Limitations of classical welfare analysis.5 Social choice theory.

There’s quite a bit of mathematics in this material...





Perfect Competition

1 problem solving exercise (20 %) + 1 in-class test (20 %) + atwo hour examination (60 %) = 100 %





Perfect Competition

I ∈ N is the number of consumers, we always take I ≥ 2 (twoor more consumers).1 I = {1, . . . , I} is the set of consumers.

n ∈ N is the number of goods, so {1, . . . , n} is the set ofgoods’ indices.

ei = (e i1, . . . , eni ) ∈ Rn

+ is consumer i ’s endowment vector.This is also commonly referred to as the vector of initialresources or initial endowment vector.

1N denotes the set of natural numbers, i.e., N = {1, 2, 3, . . .}.Martin K. Jensen (U. B’ham) Econ 320B, Set 1




Perfect Competition



ei = (e i1, . . . , eni ) ∈ Rn






Perfect Competition



ei = (e i1, . . . , eni ) ∈ Rn






Perfect Competition

e = (e1, . . . , eI ) ∈ RnI+ is the economy’s endowment vector.

xi = (x i1, . . . , x

in) ∈ Rn

+ is a (consumption) bundle forconsumer i . Collecting the I consumers’ bundles we get anallocation x = (x1, . . . , xI ) ∈ RnI

+ .

The preference relation of consumer i is denoted by �i .

The utility function of consumer i is denoted by ui .





Perfect Competition


xi = (x i1, . . . , x

in) ∈ Rn


+ .







Perfect Competition


xi = (x i1, . . . , x

in) ∈ Rn


+ .







Perfect Competition


xi = (x i1, . . . , x

in) ∈ Rn


+ .







Perfect Competition

Definition

The utility function ui : Rn+ → R represents the preference

relation �i if xi �i x̃i ⇔ ui (xi ) ≥ ui (x̃i ), xi , x̃i ∈ Rn+.





Perfect Competition

In a perfectly competitive exchange economy (with privateownership=no government), consumers sell their endowmentsat the prevailing (market) prices p = (p1, . . . , pn), pm > 0 allm, and use the resulting income,

pei =n∑

k=1

pke ik ,

to buy a consumption bundle xi ∈ Rn+, the expenditure of

which will be:

pxi =n∑

k=1

pkx ik





Perfect Competition

We always require that xi ∈ Rn+.

The bundle xi is affordable (or feasible) if:

pxi ≤ pei (1)





Perfect Competition

We always require that xi ∈ Rn+.

The bundle xi is affordable (or feasible) if:

pxi ≤ pei (1)





Perfect Competition

Consumers’ preferences are represented by utility functionsui : Rn

+ → R

Objective of the consumer:

max ui (x i1, . . . , x

in)

s.t.

{ ∑k pkx i

k ≤∑

k pke ikx ik ≥ 0 for k = 1, . . . , n

(2)





Perfect Competition

Consumers’ preferences are represented by utility functionsui : Rn

+ → RObjective of the consumer:

max ui (x i1, . . . , x

in)

s.t.

{ ∑k pkx i

k ≤∑

k pke ikx ik ≥ 0 for k = 1, . . . , n

(2)





Perfect Competition

Assumption 5.1. (JR p.188) The utility functionui : Rn

+ → R is continuous, strongly increasing, and strictlyquasi-concave.

Let’s look at each in turn...

Definition

A utility function ui is continuous if for any convergent sequence(xp)∞p=1, xp → x as p →∞, it holds that ui (xp)→ ui (x) asp →∞.





Perfect Competition




Definition






Perfect Competition




Definition






Perfect Competition

Definition

A utility function ui is strongly increasing if x ik ≥ x̃ i

k for allk = 1, . . . , n with a least one strict inequality, implies thatui (xi ) > ui (x̃i ). In words, adding more of one, or more of thegoods makes the consumer strictly better off.

When the utility function is differentiable, a sufficientcondition for it to be strongly increasing is that the partialderivatives are strictly positive.





Perfect Competition

Definition

A utility function ui is strongly increasing if x ik ≥ x̃ i

k for allk = 1, . . . , n with a least one strict inequality, implies thatui (xi ) > ui (x̃i ). In words, adding more of one, or more of thegoods makes the consumer strictly better off.

When the utility function is differentiable, a sufficientcondition for it to be strongly increasing is that the partialderivatives are strictly positive.





Perfect Competition

Definition

A utility function ui is strictly quasi-concave if the “better sets”are all strictly convex, i.e., if each of the setsB(xi ) ≡ {y ∈ Rn

+ : ui (y) ≥ ui (xi )}, where xi ∈ Rn+, is strictly

convex.

A set B ⊆ Rn is strictly convex if for every pair of elements,x , x̃ ∈ B, x 6= x̃ , and any λ ∈ (0, 1), the convex combinationλx + (1− λ)x̃ lies in the interior of B. The set is convex if forevery pair of elements, x , x̃ ∈ B, x 6= x̃ , and any λ ∈ [0, 1],the convex combination λx + (1− λ)x̃ lies in B. It is clearthat a strictly convex set is also convex.





Perfect Competition

Definition

A utility function ui is strictly quasi-concave if the “better sets”are all strictly convex, i.e., if each of the setsB(xi ) ≡ {y ∈ Rn

+ : ui (y) ≥ ui (xi )}, where xi ∈ Rn+, is strictly

convex.

A set B ⊆ Rn is strictly convex if for every pair of elements,x , x̃ ∈ B, x 6= x̃ , and any λ ∈ (0, 1), the convex combinationλx + (1− λ)x̃ lies in the interior of B. The set is convex if forevery pair of elements, x , x̃ ∈ B, x 6= x̃ , and any λ ∈ [0, 1],the convex combination λx + (1− λ)x̃ lies in B. It is clearthat a strictly convex set is also convex.


Econ 320B, Set 1 - University of Birminghamsocscistaff.bham.ac.uk/jensen/320b-Set-1-beamer.pdf · Outline Practical Information Course Overview Course Overview Method of Assessment

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