Applied general equilibrium models:Theoretical part

M.A. Keyzer

and

C.F.A. van Wesenbeeck

Overview of course

1. Introduction (Chapter 1,2, (3))

2. Applied general equilibrium: formats: (Chapter 3)

3. Taxes, tariffs and quota: (Chapter 5)

4. Dynamics: (Chapter 8)

5. Externalities: (Chapter 9)

Study load and prerequisites

• The workload of this course is 2 credits (80 hrs)– 10 hrs attending lectures

– 35 hrs preparing for lectures

– 35 hrs preparing for the exam

• Prerequisites– Knowledge of microeconomics at the level of the core course

"microeconomics" is assumed.

Lecture 1: Introduction

• Competitive equilibrium

• Negishi theorem

• Production

• Reforms

• Represent migration

• Mathematical “work horses”

• Literature: chapters: 1, 2 and first part of 3

• Sheets available at www.sow.vu.nl/downloadables.htm

Aim of lecture 1

• Highlighting the normative relevance of the competitive equilibrium model

• Showing the relation between competitive equilibrium and welfare optimality

Competitive equilibrium for an exchange economy

Competitive equilibrium for an exchange economy:

consumers are indexed commodities are indexed  consumers have utility functions (utility functions are concave increasing)

where is consumption vector with elements  for given commodity endowments determine the market clearing prices

i 1,...,mk 1,...,r

i iu ( x )

ix ikx

ip

Competitive equilibrium (continued)

Formally,

solves consumer problem:

for given income

The market clearing condition is:

and solves for equilibrium prices .

ix ( p )

ix 0 i i i imax { u ( x )| px h }

i ih p

i ii ix

p

Negishi theorem

• This competitive equilibrium can be represented (in Negishi format) as a welfare program with welfare weights adjusted to meet individual budgets.

• Both representations are equivalent!

Negishi theorem (continued)

Competitive equilibrium in Negishi format:

(a) Welfare program

where is Lagrange multiplier.

(b) Adjust welfare weights so as to satisfy budgets for every i.

i ix 0,all i i i i

i ii i

max u ( x )

subject to

x (p)

p

ii ipx p

Negishi theorem (continued)

Check equivalence between both representations:

Welfare program: first-order condition with respect to demand for commodity :

with equality if ,

which, for such that gives f.o.c. of consumer problems

k

ii k

ik

up

x

ikx 0

i i1 / i ipx p

Relation to First Welfare Theorem

“A competitive equilibrium is Pareto efficient”.

Here: Competitive equilibrium is welfare optimum.

Welfare optimum is Pareto efficient.

Requirements for proof First Welfare Theorem

• Production set has to be compact and nonempty– Note that convexity is not required

• Utility function has to be continuous and non-satiated– Note that concavity is not required

– Note that function does not have to be increasing in all commodities

Relation to Second Welfare Theorem:

“Every Pareto efficient allocation is implementable as a competitive equilibrium with transfers” .

Here: for fixed weights ,welfare program gives a competitive equilibrium with transfers equal to budget deficits:

Requirements for proof of Second Welfare Theorem are stronger than that for First Welfare Theorem

i

i i iT px p

Production

i j ix 0,all i,y ,all j i i i

i i ji i j

j j

max u ( x )

subject to

x + y (p)

y Y

where is Lagrange multiplier, and with such that budgets hold for every i.

pi i ij jpx p py

i

Production (continued)

• The production set is compact, convex, and has possibility of inaction

• Producers maximize profits under technology constraint:

max

jy j

j j

pysubject to

y Y

jY

Welfare gains from reforms

Example: the elimination of a consumer subsidy

Consumer subsidy

Consumer price

Consumer problem

for income

Market clearing

cp p

i

cx 0 i i i imax { u ( x )| p x h }

ii i i ih p T , with T x

i ii ix

Welfare gains from reforms (continued)

• Generally: subsidies, tariffs, monopoly premiums and wage subsidies can be represented by separate terms in objective with a weight factor

• Consumer welfare rises as factor is reduced

Welfare gains from reforms (continued)

i i ui(xi)

i xi

Consumer subsidy in welfare program

• Negishi program:

• Check equivalence with excess demand format

• Gain from reform: rises as is reduced

i i ix 0,all i i i i i

i ii i

max u ( x ) x

subject to

x (p)

i i i iu ( x )

• Welfare program with fixed assignment of population: people in class :

With , i.e. per capita consumption

• Adjust welfare weights so as to satisfy budgets for every group i.

Representing migration

i ix 0,all i i i i i

i ii i i i

max n u ( x )

subject to

n x n (p),

iin

i i ix X n

Representing migration (continued)

Flexible allocation of people over classes i:

where N is the given number of people in total

i i in 0;x 0,all i i i i i

i ii i i i

i i

max n u ( x )

subject to

n x n (p)

n =N

Representing migration (continued)

• This shows welfare gain from perfectly free migration

• Note that is the probability of individual

ending up in state

• Full specialization in best state

• Representation is too simple:

e.g. labor endowments are produced with commodities

. This topic is taken up in lecture 5.

i iP n / N

i

i i( x )

Mathematical “work horses”

• Key propositions– 1.4 existence of a general competitive equilibrium

– 1.5: first welfare theorem

– 1.8: representing Pareto-efficient allocation by welfare optimum

– 1.10: second welfare theorem

– 2.14: properties of the welfare optimum

– 3.1: Negishi theorem

Mathematical “work horses”(continued)

• Existence of an optimum– Convex optimizations

• Concavity of objective• Non-emptiness and convexity of constraint set• Slater’s constraint qualification for existence of shadow prices

• Characterization of the optimum (dependence of the optimum on parameters) – Maximum theorem, perturbation theorem, envelope

theorem

• Fixed point: Theorems Kakutani and Brouwer

Mapping from theorems to “work horses”1.4 Existence comp.eq.

1.8 P.E by W.O 2.14 Properties W.O 3.1 Negishi Theorem

Concavity objective

Assumptions on u(x)

Assumptions on u(x)

Assumptions on u(x) Assumptions on u(x)

Slater Constraint set of consumer optimization

Constraint set of welfare program

Constraint set of welfare program

Constraint set of welfare program

Kakutani Fixed point in prices

Fixed point in welfare weights and prices

Maximum Theorem

Continuity of consumer demand function

Continuity of consumer demand in welfare weights and endowments

Continuity of consumer demand in welfare weights and endowments; upper semicontinuity of price correspondence in welfare weights; compactness of set of prices

Envelope Theorem

Utility is partial derivative of value function w.r.t welfare weight

Perturbation theorem

Convexity and non-decreasingness of the value function in welfare weights, concavity in endowments