Exchange clear Pure exchange: model where all of the Teemu Olvio - - PDF document

SLIDE 1

1

S ystems

Analysis Laboratory

Helsinki University of Technology Session 1 - Student presentation Seminar on Microeconomics - Fall 1998 / 1

Mat-2.142 Seminar on optimization Microeconomics Session 13:

Exchange

Teemu Olvio Kari Vatanen

S ystems

Analysis Laboratory

Helsinki University of Technology Session 2 - Student presentation Seminar on Microeconomics - Fall 1998 / 2

• Partial equilibrium:

prices assumed to remain fixed (except studied one)

• General equilibrium:

all prices are variable and all markets clear Pure exchange: model where all of the economic agents are consumers

S ystems

Analysis Laboratory

Helsinki University of Technology Session 3 - Student presentation Seminar on Microeconomics - Fall 1998 / 3

• Agent i’s consumption bundle:
• Walrasian equilibrium:

xi xi xi k =( ,..., ) 1 i xi p p i i i ∑ ≤∑ ( *, * ) ω ω S ystems

Analysis Laboratory

Helsinki University of Technology Session 4 - Student presentation Seminar on Microeconomics - Fall 1998 / 4

• The aggregate excess demand

function:

• Walras’s law: For any price vector p,

we have pz(p)≡ 0; i.e., the value of the excess demand is identically zero.

z p xi p p i i i n ( ) [ ( , ) ] = − = ∑ ω ω 1 S ystems

Analysis Laboratory

Helsinki University of Technology Session 5 - Student presentation Seminar on Microeconomics - Fall 1998 / 5

• Market clearing: If demand equals

supply in k-1 markets, and , then demand must equal supply in the market.

• Free goods: If p* is a Walrasian

equilibrium and (p*) ≤ 0

0, then .

That is, if some good is in excess supply at a Walrasian equilibrium it must be a free good.

pk >0 kth zj pj *=0 S ystems

Analysis Laboratory

Helsinki University of Technology Session 6 - Student presentation Seminar on Microeconomics - Fall 1998 / 6

• Desirability: If , then

for i=1,…,k.

• Equality of demand and supply: If all

goods are desirable and p* is a Walrasian equilibrium, then z(p*)=0.

pi =0 zi p ( )>0

SLIDE 2

2

S ystems

Analysis Laboratory

Helsinki University of Technology Session 7 - Student presentation Seminar on Microeconomics - Fall 1998 / 7

• Brouwer fixed-point theorem: If f :

is a continuous function from the unit simplex to itself, there is some x in such that x=f(x).

Sk Sk − → − 1 1 Sk−1 S ystems

Analysis Laboratory

Helsinki University of Technology Session 8 - Student presentation Seminar on Microeconomics - Fall 1998 / 8

• Existence of Walrasian equilibria: If

z: is a continuous function that satisfies Walras’ law, pz(p) ≡ 0, then there exists some p* in such that z(p*) ≤ 0.

Sk Rk − → 1 Sk−1 S ystems

Analysis Laboratory

Helsinki University of Technology Session 9 - Student presentation Seminar on Microeconomics - Fall 1998 / 9

Definitions of Pareto efficiency

• Weak Pareto efficiency

– A feasible allocation x is weakly Pareto efficient allocation if there is no feasible allocation x’ such that all agents strictly prefer x’ to x

• Strong Pareto efficiency

– There is no feasible allocation x’ such that all agents weakly prefer x’ to x and some agent strictly prefer x’ to x

S ystems

Analysis Laboratory

Helsinki University of Technology Session 10 - Student presentation Seminar on Microeconomics - Fall 1998 / 10

Equivalence of weak and strong Pareto efficiency

• If preferences are continuous and

monotonic, then an allocation is weakly Pareto efficient if and only if it is strongly Pareto efficient

S ystems

Analysis Laboratory

Helsinki University of Technology Session 11 - Student presentation Seminar on Microeconomics - Fall 1998 / 11

Edgeworth box

Consumer 1 Consumer 2 good 1 good 2

max u1(x1)

x1, x2

such that u2(x2) û2 x1 + x2 = ω ω1 + ω ω2 ≥

S ystems

Analysis Laboratory

Helsinki University of Technology Session 12 - Student presentation Seminar on Microeconomics - Fall 1998 / 12

Walrasian equilibrium

• An allocation-price pair (x, p) is a

Walrasian equilibrium if

– the allocation is feasible ∑ ∑ xi = ∑ ∑ ω ωi – each agent is making an optimal choice from his budget set

• If xi’ is preferred by agent i to xi, then

pxi’ > pω ωi

SLIDE 3

3

S ystems

Analysis Laboratory

Helsinki University of Technology Session 13 - Student presentation Seminar on Microeconomics - Fall 1998 / 13

First Theorem of Welfare Economics

• If (x, p) is a Walrasian equilibrium, then x

is Pareto efficient

S ystems

Analysis Laboratory

Helsinki University of Technology Session 14 - Student presentation Seminar on Microeconomics - Fall 1998 / 14

Second Theorem of Welfare Economics

• x* is a Walrasian equilibrium for the initial

endowments ω

ωi = xi* for i = 1, … , n

– if x* is a Pareto efficient allocation in which each agent holds a positive amount of each good and if preferences are convex, continuous and monotonic

• If a competitive equilibrium (p, x’) exists

from the initial endowments ω

ωi = xi*, then,

in fact, (p, x*) is a competitive equilibrium

S ystems

Analysis Laboratory

Helsinki University of Technology Session 15 - Student presentation Seminar on Microeconomics - Fall 1998 / 15

Calculus

• Market equilibrium

– If (x*, p*) is a market equilibrium with each consumer holding a positive amount of every good, then there exist a set of numbers (λ1, …, λn) such that:

Dui(x*) = λip* i = 1, …, n

S ystems

Analysis Laboratory

Helsinki University of Technology Session 16 - Student presentation Seminar on Microeconomics - Fall 1998 / 16

Calculus

• Pareto efficiency

– A feasible allocation x* is Pareto efficient if and only if x* solves the following n maximization problems for i = 1, …, n max ui(xi)

(xi

g)

such that ∑

∑ xig ω

ωg g = 1, …, k uj(xj*) uj(xj) j ≠ i

n

i=1

≤ ≤

S ystems

Analysis Laboratory

Helsinki University of Technology Session 17 - Student presentation Seminar on Microeconomics - Fall 1998 / 17

Social welfare function

• Social utility W(u1,…, un)
• If x* maximizes a sosial welfare function,

then x* is Pareto efficient

max W(u1(x1),…, un(xn)) such that ∑

∑ xig ω

ωg g = 1, …, k

n

i=1

≤

S ystems

Analysis Laboratory

Helsinki University of Technology Session 18 - Student presentation Seminar on Microeconomics - Fall 1998 / 18

Welfare maximization

• If x* is Pareto efficient allocation (xi*>>0)
• If utility functions are concave, continuous

and monotonic functions

• Then there is some choice of weights ai*

such that x* maximizes ∑ ∑ ai* ui(xi) subject to resource constraints

– now ai* = 1/ λi where λi is the ith agent’s marginal utility of income

SLIDE 4

4

S ystems

Analysis Laboratory

Helsinki University of Technology Session 19 - Student presentation Seminar on Microeconomics - Fall 1998 / 19

Relationships

• Competitive equilibria are always Pareto

efficient

• Pareto efficient allocations are competitive

equilibria under convexity assumptions and endowment redistribution

• Welfare maxima are always pareto efficient
• Pareto efficient allocations are welfare maxima

under concavity assumptions for some choice

• f welfare weights