advanced microeconomics preliminary remarks · advanced microeconomics preliminary remarks...

Advanced Microeconomics

Preliminary Remarks

Ronald Wendner

Department of EconomicsUniversity of Graz, Austria

Course # 320.911

Todays program

� Organization of course

� Important principles of notation we gonna use

� Preference and choice

R. Wendner (U Graz, Austria) Microeconomics 2 / 15

Organization

Organization of our course

� Essential prerequisites

320.901 Mathematics

320.902 Game Theory

Familiarity with the basic concepts in microeconomics as offered byundergraduate microeconomics textbooks such as Nicholson, MicroeconomicTheory

Organization

� Required reading

Mas-Colell, A., M.D. Whinston, J.R. Green (1995), Microeconomic Theory,New York, Oxford: Oxford University Press

My presentation + additional materials:My website → Teaching → Advanced Microeconomics

I my materials are no substitute

Organization

Further references

Sydsaeter, K., P. Hammond (2012, 4th ed.), Essential Mathematics forEconomic Analysis, Harlow: Pearson Education Ltd.

Corbae, D., M.B. Stinchcombe, J. Zeman (2009), An Introduction toMathematical Analysis for Economic Theory and Econometrics, Princeton etal.: Princeton University Press.

Sydsaeter, K., P. Hammond, A. Seierstad, A. Strom (2008, 2nd ed.), FurtherMathematics for Economic Analysis, Harlow: Pearson Education Ltd.

Novshek, W. (1993), Mathematics for Economists, San Diego et al.: AcademicPress Inc.

Velleman, D.J. (2006), How to Prove it, Cambridge et al.: CambridgeUniversity Press

Dixit, A. (1990), Optimization in Economic Theory, New York: OxfordUniversity Press.

put on reserve (“Semesterhandapparat”) in FB.

Organization

� Topics

Notation

Preference and choice

Consumer choice

Classical demand theory

Aggregate demand

Production

Equilibrium and basic welfare properties

Organization

� Typical agenda & workload

recap (last classes’ main results)

lecture or problem sets

required reading for following class

workload for weeks with 1 class per week:

6 ECTS = 6 x 25 hours = 150h = 10h per week = 8h per weekin addition to class

Organization

� Typical agenda & workload

recap (last classes’ main results)

lecture or problem sets

required reading for following class

workload for weeks with 1 class per week:

6 ECTS = 6 x 25 hours = 150h = 10h per week = 8h per weekin addition to class

Organization

� Grading based on percentage grades

in class participation: 15 %

midterm exam, 24 April, 2018: 40%

final exam, 26 June, 2018: 45%

I letter grades:

86% - 100%: Sehr gut (A);73% - 85%: Gut (B);60% - 72%: Befriedigend (C);50% - 59%: Genügend (D);0% - 49%: Nicht genügend (F).

Organization

� Grading based on percentage grades

in class participation: 15 %

midterm exam, 24 April, 2018: 40%

final exam, 26 June, 2018: 45%

I letter grades:

86% - 100%: Sehr gut (A);73% - 85%: Gut (B);60% - 72%: Befriedigend (C);50% - 59%: Genügend (D);0% - 49%: Nicht genügend (F).

Notation

Vector- and Matrix Notation

Mathematical ingredients in microeconomics

functions, (binary) relations, correspondences

scalars, vectors, matrices

frequently RN ≡ ×Ni=1 R = R× R× ...× R

(= N-fold Cartesian product)

x ∈ RN =

x1x2...

is a column (!) vector

with transpose xT = (x1, x2, . . . , xN )

Notation

Vector- and Matrix Notation

Mathematical ingredients in microeconomics

functions, (binary) relations, correspondences

scalars, vectors, matrices

frequently RN ≡ ×Ni=1 R = R× R× ...× R

(= N-fold Cartesian product)

x ∈ RN =

x1x2...

is a column (!) vector

with transpose xT = (x1, x2, . . . , xN )

Notation

Consider vectors x, y ∈ RN

inner product x · y ≡ xT y

x ≥ 0⇔ xi ≥ 0, i = 1, ...N ⇔ x ∈ RN+

x = 0⇔ xi = 0, i = 1, ...N

x� 0⇔ xi > 0, i = 1, ...N ⇔ x ∈ RN++

single- and vector valued functions

single valued functionf(x) : RN → R, with x ∈ RN , N ≥ 1

vector valued functionf(x) : RN → RM, with x ∈ RN , N ≥ 1, M ≥ 1

vector of M functions, each of which is defined on the domain RN

Notation

x ≥ 0⇔ xi ≥ 0, i = 1, ...N ⇔ x ∈ RN+

x = 0⇔ xi = 0, i = 1, ...N

x� 0⇔ xi > 0, i = 1, ...N ⇔ x ∈ RN++

Notation

x ≥ 0⇔ xi ≥ 0, i = 1, ...N ⇔ x ∈ RN+

x = 0⇔ xi = 0, i = 1, ...N

x� 0⇔ xi > 0, i = 1, ...N ⇔ x ∈ RN++

Notation

x ≥ 0⇔ xi ≥ 0, i = 1, ...N ⇔ x ∈ RN+

x = 0⇔ xi = 0, i = 1, ...N

x� 0⇔ xi > 0, i = 1, ...N ⇔ x ∈ RN++

Notation

x ≥ 0⇔ xi ≥ 0, i = 1, ...N ⇔ x ∈ RN+

x = 0⇔ xi = 0, i = 1, ...N

x� 0⇔ xi > 0, i = 1, ...N ⇔ x ∈ RN++

Notation

Consider (single function) f(x) : RN → R, with x ∈ RN , N ≥ 1

Gradient at x̄: ∇f(x̄) ∈ RN (column vector)

∇f(x̄) ≡

∂ f(x̄)/∂ x1∂ f(x̄)/∂ x2

...∂ f(x̄)/∂ xN

What is the gradient of a utility/production function?

Notation

Consider (single function) f(x) : RN → R, with x ∈ RN , N ≥ 1

Gradient at x̄: ∇f(x̄) ∈ RN (column vector)

∇f(x̄) ≡

∂ f(x̄)/∂ x1∂ f(x̄)/∂ x2

...∂ f(x̄)/∂ xN

What is the gradient of a utility/production function?

Notation

Consider (the vector-valued function) f(x) : RN → RM, with

x ∈ RN , N ≥ 1, M ≥ 1, and f(x) =

f1(x)f2(x)

...fM (x)

Jacobian matrix at x̄: Df(x̄) is M ×N matrix of FO partial derivatives

Df(x̄) ≡

∂ f1(x̄)/∂ x1, ∂ f1(x̄)/∂ x2, . . . , ∂ f1(x̄)/∂ xN

∂ f2(x̄)/∂ x1, ∂ f2(x̄)/∂ x2, . . . , ∂ f2(x̄)/∂ xN

...∂ fM (x̄)/∂ x1, ∂ fM (x̄)/∂ x2, . . . , ∂ fM (x̄)/∂ xN

for M = 1: Df(x) = [∇f(x)]T

Notation

x ∈ RN , N ≥ 1, M ≥ 1, and f(x) =

f1(x)f2(x)

...fM (x)

Df(x̄) ≡

∂ f1(x̄)/∂ x1, ∂ f1(x̄)/∂ x2, . . . , ∂ f1(x̄)/∂ xN

∂ f2(x̄)/∂ x1, ∂ f2(x̄)/∂ x2, . . . , ∂ f2(x̄)/∂ xN

...∂ fM (x̄)/∂ x1, ∂ fM (x̄)/∂ x2, . . . , ∂ fM (x̄)/∂ xN

for M = 1: Df(x) = [∇f(x)]T

Notation

x ∈ RN , N ≥ 1, M ≥ 1, and f(x) =

f1(x)f2(x)

...fM (x)

Df(x̄) ≡

∂ f1(x̄)/∂ x1, ∂ f1(x̄)/∂ x2, . . . , ∂ f1(x̄)/∂ xN

∂ f2(x̄)/∂ x1, ∂ f2(x̄)/∂ x2, . . . , ∂ f2(x̄)/∂ xN

...∂ fM (x̄)/∂ x1, ∂ fM (x̄)/∂ x2, . . . , ∂ fM (x̄)/∂ xN

for M = 1: Df(x) = [∇f(x)]T

Notation

Restrictions on the Jacobian

consider f(x, y) with x ∈ RK , y ∈ RL so that f(x, y) : RK+L → RM

with y ∈ RL being constant

Dxf(x, y) is a Jacobian of dimension M ×K

Consider single function g(x) : RN → R

Hessian matrix D2g(x) is the symmetric N ×N matrix

D2g(x) = D(∇g(x)) =∂ g(x)/(∂ x1∂ x1), ∂ g(x)/(∂ x1∂ x2), . . . , ∂ g(x)/(∂ x1∂ xN )∂ g(x)/(∂ x2∂ x1), ∂ g(x)/(∂ x2∂ x2), . . . , ∂ g(x)/(∂ x2∂ xN )

...∂ g(x)/(∂ xN ∂ x1), ∂ g(x)/(∂ xN ∂ x2), . . . , ∂ g(x)/(∂ xN ∂ xN )

Notation

� The beauty of matrix notation

I remember the chain rule for M=N=1:

g(x) : R→ R, and f(.) : R→ R so that f(g(x)) : R→ Rthen: ∂ [f(g(x))]/∂ x = f ′(g(x)).g′(x)

or using our differential operator D:Dxf(g(x)) = Df(x).Dg(x)

lets go fully general ...

Notation

Chain rule for M ≥ 1, N ≥ 1

consider f(g(x)), where f : RK → RM , g : RN → RK , x ∈ RN :

f(g(x)) =

f1(g1(x), ..., gK(x))f2(g1(x), ..., gK(x))...

fM (g1(x), ..., gK(x))

what is Dxf(g(x)) ?

Dxf(g(x)) = Df(g(x)) ·Dg(x)

dimensions: Dxf(g(x))[M×N ] = Df(g(x))[M×K] ·Dg(x)[K×N ]

Notation

f(g(x)) =

f1(g1(x), ..., gK(x))f2(g1(x), ..., gK(x))...

fM (g1(x), ..., gK(x))

what is Dxf(g(x)) ?

Notation

f(g(x)) =

f1(g1(x), ..., gK(x))f2(g1(x), ..., gK(x))...

fM (g1(x), ..., gK(x))

what is Dxf(g(x)) ?

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