MICROECONOMICS II.

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MICROECONOMICS II.

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ELTE Faculty of Social Sciences, Department of Economics

Microeconomics II.

week 3

GENERAL EQUILIBRIUM THEORY, PART 2 Author: Gergely K®hegyi

Supervised by Gergely K®hegyi

February 2011

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week 3 Gergely K®hegyi

Equilibrium with production

One participant, two goods, one factor of production Two participants, two goods, one factor of production Two consumers, two producers, two goods, one factor of production N goods, M consumers, R producers and K factors of production Main questions Imperfect markets

Prepared by: Gergely K®hegyi, using Jack Hirshleifer, Amihai Glazer és David Hirshleifer (2009) Mikroökonómia. Budapest:

Osiris Kiadó, ELTECON-könyvek (henceforth: HGH), and Kertesi Gábor (ed.) (2004) Mikroökonómia el®adásvázlatok.

http://econ.core.hu/ kertesi/kertesimikro/ (henceforth: KG).

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Draft

1 Equilibrium with production

One participant, two goods, one factor of production Two participants, two goods, one factor of production Two consumers, two producers, two goods, one factor of production

N goods, M consumers, R producers and K factors of production

2 Main questions

3 Imperfect markets

(7)

First example

Robinson consumes sh and coconut, which he produces with labor.

x K : Robinson's coconut consumption

x _H : Robinson sh consumption

h: factor of production (working hours): ¯ h = 10 Prod. functions (constant return to scale):

x

H

= 10h

H

x

K

= 20h

K

Resource constraint:

h H + h K = 10

x _H 10 + x _K

20 = 10

x K = 200 − 2x H

(8)

First example (cont.)

Production-possibility set:

x _K + 2x _H ≤ 200 x _K , x _H ≥ 0 Production-possibility frontier:

x K + 2x H = 200 Marginal rate of transformation:

MRT = dx k

dx _H = − 2

(9)

Second example

Robinson consumes sh and coconut, which he produces with labor.

x _K : Robinson's coconut consumption

x _H : Robinson sh consumption

h: factor of production (working hours): ¯ h = 10 Prod. functions (decreasing return to scale):

x

H

= √ h

H

x

K

= √ h

K

Resource constraint:

h _H + h _K = 25

x _H ² + x _K ² = 25

(10)

Second example (cont.)

Production-possibility set:

x _K ² + x _H ² ≤ 25 x K , x H ≥ 0 Production-possibility frontier (PPF):

x _K ² + x _H ² = 25

Transformation curve (implicit form of the PPF curve):

T ( x _H , x _K ) = 0

T ( x H , x K ) = x _K ² + x _H ² − 25

(11)

Deriving the marginal rate of transformation from the transformation curve

T ( x H , x K ) = 0

Total dierential:

dT ( x _H , x _K ) = ∂ T

∂ x H dx _H + ∂ T

∂ x K dx _K Along the curve dT ( x _H , x _K ) = 0

MRT = dx K

dx _H = − ∂ T /∂ x H

∂ T /∂ x _K E.g.:

MRT = − 2x H

2x _K = − x H

x _K

(12)

Deriving the marginal rate of transformation from

the transformation curve (cont.)

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Relation of the marginal rate of transformation and the marginal product

x H = f H ( h H ) x K = f K ( h K )

h _H = f _H ⁻ ¹ ( x _H ) h _K = f _K ⁻ ¹ ( x _K ) h _H + h _K = ¯ h T ( x _H , x _K ) = f _H ⁻ ¹ ( x _H ) + f _K ⁻ ¹ ( x _K ) − ¯ h

MRT = − ∂ T /∂ x H

∂ T /∂ x K = − df _H ⁻ ¹ / dx _H df _K ⁻ ¹ / dx _K

Since ^dh _dx = _dh ¹ _/ _dx (can be proven mathematically precisely), then

df

⁻¹

dx = _mp ¹ . Thus

MRT = − 1 / mp _H

1 / mp K = − mp _K

mp H

(14)

Role of the social planner

Maximize: U ( x ₁ , x ₂ ) → max _x

₁

, x

2

Subject to: T ( x 1 , x 2 ) = 0

Lagrange-function: L = U ( x ₁ , x ₂ ) − λ T ( x ₁ , x ₂ )

∂L

∂x1

=

_∂^∂_x^U

1

− λ

_∂^∂_x^T

1

= 0

∂L

∂x2

=

_∂^∂U_x

2

− λ

_∂^∂T_x

2

= 0

MRS = MRT

(15)

Decentralized decisions

Robinson as producer (produces two goods with one factor):

Fix cost of production : F , price of production factor (labor) Maximize: Π = p

1

y

1

+ p

2

y

2

− F → max

y1,y2

Subject to: T ( y

1

, y

2

) = 0

Lagrange-function: L = p

1

y

1

+ p

2

y

2

− F − λT (y

1

, y

2

) First order condition:

∂L

∂y1

= p

1

− λ

_∂^∂_y^T

1

= 0

∂L

∂y2

= p

2

− λ

_∂^∂T_y

2

= 0

MRT = − p

1

p

2

(16)

Decentralized decisions (cont.)

Robinson as consumer (income: income from labor (F )+

capital income ( Π )):

Maximize: U ( x

1

, x

2

) → max

x1,x2

Subject to: p

1

x

1

+ p

2

x

2

= F + Π

Lagrange-function: L = U ( x

1

, x

2

) − λ( p

1

x

1

+ p

2

x

2

− F − Π) First order condition:

∂L

∂x1

=

_∂^∂U_x

1

− λ p

1

= 0

∂L

∂x2

=

_∂x^∂^U

2

− λ p

2

= 0

MRS = − p

1

p

2

MRS = MRT Equilibrium conditions:

y ₁ ( p ₁ , p ₂ ) = x ₁ ( p ₁ , p ₂ ); y ₂ ( p ₁ , p ₂ ) = x ₂ ( p ₁ , p ₂ )

(17)

Draft

1 Equilibrium with production

One participant, two goods, one factor of production Two participants, two goods, one factor of production Two consumers, two producers, two goods, one factor of production

N goods, M consumers, R producers and K factors of production

2 Main questions

3 Imperfect markets

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Notation

x

_K^R

: Robinson's consumption of coconut

x

_H^R

: Robinson's consumption of sh

h

^R

: factor of production ( h ¯

^R

= 10)

Production functions:

x

_H^R

= 10h

_H^R

x

_K^R

= 20h

_K^R

Resource constraint:

h

^R_H

+ h

^R_K

= 10

Production-possibility set:

x

_H^P

+ 2x

_K^P

≤ 200 x

_H^P

, x

_K^P

≥ 0

x

_K^P

: Friday's coconut consumption

x

_H^P

: Friday's sh consumption h

^P

: factor of production ( ¯ h

^P

= 10)

Production functions:

x

_H^P

= 20h

^P_H

x

_K^P

= 10h

^P_K

Resource constraint:

h

^P_H

+ h

^P_K

= 10

Production-possibility set:

2x

_H^R

+ x

_K^R

≤ 200

x

_H^R

, x

_K^R

≥ 0

(19)

Production-possibility frontier

x

_H^R

10 + x

_K^R

20 = 10 x

_K^R

= 200 − 2x

_H^R

MRT

^R

= −2

x

_H^P

20 + x

_K^P

10 = 10 x

_K^P

= 200 − 0 , 5x

_H^P

MRT

^P

= −0, 5

Robinson has comparative advantage in coconut production, and

Friday in sh production.

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Utility of exchange

The utility of exchange comes from the division of labor.

(21)

Role of the social planner

Searching for Pareto-ecient allocations:

Maximize: U A ( x ₁ ^A , x ₂ ^A ) → max _x

₁^A

, x

₂^A

, x

₁^B

, x

₂^B

Subject to:

U

B

( x

1^B

, x

2^B

) = ¯ U

B

T ( x

1

, x

2

) = 0 x

1

= x

₁^A

+ x

₁^B

x

2

= x

₂^A

+ x

₂^B

Lagrange-function:

L = U _A ( x ₁ ^A , x ₂ ^A ) − λ U _B ( x ₁ ^B , x ₂ ^B ) − U ¯ _B

−

−µ T x ₁ ^A + x ₁ ^B , x ₂ ^A + x ₂ ^B

(22)

Role of the social planner (cont.)

First order conditions:

∂L

∂x₁^A

=

^∂^U^A

∂x₁^A

− µ

_∂^∂_x^T

1

= 0

∂L

∂x₂^A

=

^∂^U^A

∂x₂^A

− µ

_∂^∂_x^T

2

= 0

∂L

∂x₁^B

= −λ

^∂U_∂x_B^B

1

− µ

_∂^∂_x^T

1

= 0

∂L

∂x₂^B

= −λ

^∂U^B

∂x^B₂

− µ

_∂x^∂T

2

= 0

MRS A = MRT

MRS _B = MRT

MRS A = MRS B = MRT

(23)

Role of the social planner (cont.)

(24)

Draft

1 Equilibrium with production

One participant, two goods, one factor of production Two participants, two goods, one factor of production Two consumers, two producers, two goods, one factor of production

N goods, M consumers, R producers and K factors of production

2 Main questions

3 Imperfect markets

(25)

Decentralized decisions

The two companies (1 and 2) produces the two goods with one factor of production (labor). The two consumers (A and B) decide how much to consume besides their endowments, and how much labor to supply depending on their income, the prices of goods and the price of the factor of production. Consumers income stem from their labor (working for the companies) and as owners, from the prot of the companies.

Product prices: p ₁ , p ₂ , price of factor of production: w.

Produced quantities of products: y 1 , y 2 . Used quantities of factors of production: L ₁ , L ₂ Consumers' endowments: ω Â ₁ , ω Â ₂ , ω ₁ ^B , ω ₂ ^B Consumed quantities: x ₁ Â , x ₂ Â , x ₁ ^B , x ₂ ^B Oered labor: h _A , h _B

The share of consumer A from the prot of rm 1 is: θ _A1

Total prot of the companies are distributed among the

consumers: θ _A1 + θ _B1 = 1 , θ _A2 + θ _B2 = 1

(26)

Decentralized decisions (cont.)

Competitive mechanism I. (Optimal decision of companies):

Maximize:

π

₁

= p

1

y

1

− wL

1

→ max

y1,L1

Subject to: y

1

= f

1

( L

1

) First order condition:

p

1

mp

L1

= w Solution:

Labor demand function:

L

1

(p

1

, w ) Supply function:

y

1

( p

1

, w )

Prot function: π

₁

(p

1

, w )

Maximize:

π

₂

= p

2

y

1

− wL

2

→ max

y2,L2

Subject to: y

2

= f

2

( L

2

) First order condition:

p

2

mp

L2

= w Solution:

Labor demand function:

L

2

(p

2

, w ) Supply function:

y

2

( p

2

, w )

Prot function: π

₂

(p

2

, w )

(27)

Decentralized decisions (cont.)

Competitive mechanism II. (optimal decision of consumer): A consumer

Maximize: U

A

( x

₁^A

, x

₂^A

, h

A

) → max

_xA 1,x₂^A,hA

Subject to: p

1

x

₁^A

+ p

2

x

₂^A

= p

1

ω

^A₁

+ p

2

ω

^A₂

+ wh

A

+ θ

_A1

π

₁

+ θ

_A2

π

₂

First order condition:

MRS

₁₂^A

= −

^p_p¹

2

, MRS

_1h^A

= −

^p_w¹

, (MRS

_2h^A

= −

^p_w²

) Solution:

Labor supply function.: h

A

(p

1

, p

2

, w )

Demand function: x

₁^A

( p

1

, p

2

, w ), x

₂^A

( p

1

, p

2

, w )

(28)

Decentralized decisions (cont.)

Competitive mechanism II. (optimal decision of consumer): B consumer

Maximize: U

B

( x

₁^B

, x

₂^B

, h

B

) → max

_xB 1,x₂^B,hB

Subject to: p

1

x

₁^B

+ p

2

x

₂^B

= p

1

ω

₁^B

+ p

2

ω

^B₂

+ wh

B

+ θ

_B1

π

₁

+ θ

_B2

π

₂

First order condition:

MRS

₁₂^B

= −

^p_p¹

2

, MRS

_1h^B

= −

^p_w¹

, (MRS

_2h^B

= −

^p_w²

) Solution:

Labor demand function: h

B

(p

1

, p

2

, w )

Demand function: x

₁^B

( p

1

, p

2

, w ), x

₂^B

( p

1

, p

2

, w )

(29)

Decentralized decisions (cont.)

Competitive mechanism III. (market equilibrium conditions):

Product markets:

x ₁ Â ( p ₁ , p ₂ , w ) + x ₁ ^B ( p ₁ , p ₂ , w ) = y ₁ ( p ₁ , w ) + ω Â ₁ + ω ₁ ^B x ₂ Â ( p ₁ , p ₂ , w ) + x ₂ ^B ( p ₁ , p ₂ , w ) = y ₂ ( p ₂ , w ) + ω Â ₂ + ω ₂ ^B Factor market (labor market):

L 1 ( p 1 , w ) + L 2 ( p 2 , w ) = h A ( p 1 , p 2 , w ) + h B ( p 1 , p 2 , w )

Parameters: p ₁ ^∗ , p ₂ ^∗ , w ^∗

(30)

Decentralized decisions (cont.)

Consequence

Number of parameters and equations are equal.

Note

Since (product and factor) demand functions are zero order

homogeneous (NO MONEY ILLUSION), one of the products or

factors can be used as numeraire. Let w = ˙ 1. Thus the system of

equations seems to be over determined (more equations than

parameters).

(31)

Decentralized decisions (cont.)

Statement

Walras-law (exchange economy with production): The total value of demanded and supplied goods equals, that is the aggregate market over demand is zero (with any price system):

p 1 z 1 ( p 1 , p 2 , w ) + p 2 z 2 ( p 1 , p 2 , w ) + wz h ( p 1 , p 2 , w ) ≡ 0 , where

z

1

( p

1

, p

2

, w ) = x

1^A

( p

1

, p

2

, w ) − ω

^A₁

+ x

1^B

( p

1

, p

2

, w ) − ω

^B₁

− y

1

( p

1

, p

2

, w ) , z

2

( p

1

, p

2

, w ) = x

2^A

( p

1

, p

2

, w ) − ω

^A₂

+ x

2^B

( p

1

, p

2

, w ) − ω

^B₂

− y

2

( p

1

, p

2

, w ) and z

h

( p

1

, p

2

, w ) =

L

1

( p

1

, p

2

, w ) + L

2

( p

1

, p

2

, w ) − h

A

( p

1

, p

2

, w ) − h

B

( p

1

, p

2

, w ) .

(32)

Decentralized decisions (cont.)

Proof

Since equilibrium is based on optimal decisions, quantities and prices will satisfy consumer budget constraints. Let's add the budget constraint of the two consumer and rearrange it:

p

1

x

1^A

+ p

2

x

2^A

≡ p

1

ω

₁^A

+ p

2

ω

₂^A

+ wh

A

+ θ

_A1

π

₁

+ θ

_A2

π

₂

p

1

x

₁^B

+ p

2

x

₂^B

≡ p

1

ω

₁^B

+ p

2

ω

^B₂

+ wh

B

+ θ

_B1

π

₁

+ θ

_B2

π

₂

p

1

( x

1^A

+ x

1^B

− ω

^A₁

− ω

^B₁

) + p

2

( x

2^A

+ x

2^B

− ω

₂^A

− ω

^B₂

) ≡

≡ w (h

A

+ h

B

) + π

₁

(θ

_A1

+ θ

_B1

) + π

₂

(θ

_A2

+ θ

_B2

) since rm prot is full allocated among consumers:

p

1

( x

1^A

+ x

1^B

−ω

^A₁

− ω

₁^B

) + p

2

( x

2^A

+ x

2^B

−ω

^A₂

−ω

^B₂

) ≡ w ( h

A

+ h

B

) +π

₁

+π

₂

(33)

Decentralized decisions (cont.)

Proof

Using the denition of the rm prot:

p

1

( x

₁^A

+ x

₁^B

− ω

^A₁

− ω

^B₁

) + p

2

( x

₂^A

+ x

₂^B

− ω

₂^A

− ω

^B₂

) ≡

≡ w (h

A

+ h

B

) + p

1

y

1

− wL

1

+ p

2

y

2

− wL

2

Rearranging it:

p

1

(x

₁^A

+ x

₁^B

− ω

₁^A

− ω

₁^B

− y

1

) + p

2

(x

₂^A

+ x

₂^B

− ω

₂^A

− ω

^B₂

− y

2

)+

+w (L

1

+ L

2

− h

A

− h

B

) ≡ 0, That is:

p

1

z

1

( p

1

, p

2

, w ) + p

2

z

2

( p

1

, p

2

, w ) + wz

h

( p

1

, p

2

, w ) ≡ 0 .

(34)

Decentralized decisions (cont.)

Consequence

Due to the Walras-law the equilibrium conditions will not be independent (three equilibrium equations, two price parameters).

So one of the equations can be dropped, and the system will not

be over determined.

(35)

Algorithm for search for equilibrium with production

Algorithm

Writing up individual (producer and consumer) optimum equations

Solving the producer optimum (supply, factor demand and prot functions)

Solving the consumer optimum (demand and factor supply) Writing the market equilibrium equations (demand=supply on each market)

Choosing the numeraire good (rewriting the demand and supply functions so that they depend on the price ratio) Dening equilibrium price (one of the equilibrium equations can be dropped)

Finding the individual consumption and supply of goods.

(36)

Algorithm for search for equilibrium with production (cont.)

Note

The above algorithm can be generalized to an N product, M consumer, R rm and K factor of production system.

Példa:

Technology of two companies: y ₁ = √

L ₁ , y ₂ = √ L ₂ Utility functions of two consumers:

U _A = x ₁ ^A x ₂ ^A

h A , U _B = x ₁ ^B x ₂ ^B h B

Endowments of two consumers:

ω ₁ ^A = 100 , ω ^A ₂ = 200 , ω ₁ ^B = 300 , ω ₂ ^B = 400 , h _A = 16 , h _B = 16 (both consumers work maximum 16 hours a day)

θ _A1 = 0 , 2 ; θ _A2 = 0 , 8 ; θ _A1 = 0 , 6 ; θ _A2 = 0 , 4 (The A consumer

receives 20% of consumers 1st rm prot,... etc.)

(37)

Draft

1 Equilibrium with production

One participant, two goods, one factor of production Two participants, two goods, one factor of production Two consumers, two producers, two goods, one factor of production

N goods, M consumers, R producers and K factors of production

2 Main questions

3 Imperfect markets

(38)

General equilibrium of an N product, M consumer, R rm and K production factor economy

Parameters:

M ∗ N (N pc. consumption good, M pc. consumers) R ∗ K (K pc. factor of production, R pc. rm) N pc. consumption price

K pc. factor of production price

Number of parameters: M ∗ N + N + R ∗ K + K Equations:

M ∗ N pc. individual optimum constraint (rst order

constraints and budget constraints for the Lagrange-variables) R ∗ K pc. individual producer optimum constraint (rst order constraints+production functions+ rst order conditions for the Lagrange-variables)

N pc. equilibrium condition on the product markets:

aggregate demand = aggregate supply

K pc. equilibrium condition on the factor markets: aggregate demand = aggregate supply

Number of equations: M ∗ N + N + R ∗ K + K

So the number of equations and parameters are equal.

(39)

General equilibrium of an N product, M consumer, R rm and K production factor economy (cont.)

BUT since only relative prices matter (demand and supply functions are zero order homogeneous) a numeraire can be choosen. ( − 1 parameter).

So the system seems over determined. (More equations than parameters.)

BUT due to the Walras-law, equilibrium equations are not independent!

So the system is not over determined. Dropping one

equilibrium equation, the equilibrium can be found according to the algorithm.

Note

Counting equations might lead to wrong conclusion however.

Negative prices can turn out, since budget constraints and

equilibrium conditions are inequalities and not equalities! →

Problem of the existence of equilibrium (see below).

(40)

Fundamental theorems of the welfare state with production

Statement

The two fundamental theorems of the welfare state hold even with

production included in the system.

(41)

Fundamental theorems of the welfare state with production (cont.)

Proof

Individual consumer optimums: MRS

₁₂^A

= −

^p_p¹

2

, MRS

_1h^A

= −

^p_w¹

and MRS

₁₂^B

= −

^p_p¹

2

, MRS

_1h^B

= −

^p_w¹

. Thus

MRS

12^A

= MRS

12^B

MRS

1h^A

= MRS

1h^B

. From the production functions:

y

1

= f

1

(L

1

) y

2

= f

2

(L

2

) L

1

+ L

2

= h

A

+ h

B

L

1

= f

₁⁻¹

( y

1

) L

2

= f

₂⁻¹

( y

2

) F ( y

1

, y

2

) ˙ = f

₁⁻¹

( y

1

) + f

₂⁻¹

( y

2

) − h

A

− h

B

Transformation curve:

T (y

1

, y

2

) ˙ =F(y

1

+ ω

₁^A

+ ω

₁^B

, y

2

+ ω

₂^A

+ ω

^B₂

)

(42)

Fundamental theorems of the welfare state with production (cont.)

Proof

Producer optimum mp

1

=

_p^w

1

és mp

2

=

_p^w

2

. Thus MRT = − ∂ T /∂ y

1

∂T /∂y

2

= mp

2

mp

1

= − w / p

2

w /p

1

= − p

1

p

2

,

hence MRS

₁₂^A

= MRS

₁₂^B

= MRT , which means the the rst theorem of the welfare state holds.

Let x

1^A

, x

2^A

, h

A

, x

1^B

, x

2^B

, h

B

, L

1

, L

2

, y

1

, y

2

a random Pareto-ecient state.

Let the numeraire be w = ˙ 1. Then let's choose prices p

1

, p

2

so that:

MRS

_1h^A

= MRS

_1h^B

= ˙

^p_w¹

and MRS

₁₂^A

= MRS

₁₂^B

= MRT = ˙ −

^p_p¹

2

. Then let

us reallocate endowments so that consumer budget constraints hold

with the chosen prices. Thus the second theorem of the welfare state

holds as well.

(43)

Four main questions of the general equilibrium theory

Existence: Is there an equilibrium?

Eciency: Is the equilibrium ecient?

Uniqueness: Is the equilibrium unique, or more price systems are possible?

Stability: If the economy moves from the equilibrium (e.g.

demand or technological shock), can it return?

(44)

Existence

The existence theorem of Arrow and Debreu (1954) says that if we include some limitations to the consumer and producer side, then competitive equilibrium exists. This does not come from the equation solving procedure. Mathematical foundations of the problem are rather complicated, but the conditions for existence are the following (only a list):

Individual producers (rms) have a convex and closed production-possibility set, which included the origin. So returns to scale are not increasing and production can be shut down.

The aggregate production set does not include the positive origin, so every production includes some factor use.

Aggregate production is irreversible.

The possible consumer sets are convex, closed and limited sets.

The individual utility functions, representing preferences, are

continuous and monotonic functions.

(45)

Existence (cont.)

Indierence surfaces are convex.

Consumers have endowments.

All rm prot is allocated among consumers in a xed ratio.

(46)

Eciency

What assumptions we have to make so that the welfare theorems hold?

E.g.: Without convexity the second theorem does not hold.

(47)

Uniqueness

What are the conditions for the equilibrium to be unique? (It is

not indierent whether there is one or more price systems!)

Sidetrack:

(48)

Uniqueness (cont.)

Revealed preferences

p 1 x ₁ ^∗ + p 2 x ₂ ^∗ = p 1 ω ₁ + p 2 ω ₂ p ₁ ⁰ x ₁ ⁰ + p ₂ ⁰ x ₂ ⁰ = p ⁰ ₁ ω ₁ + p ₂ ⁰ ω ₂ p 1 x ₁ ⁰ + p 2 x ₂ ⁰ > p 1 ω ₁ + p 2 ω ₂ p ⁰ ₁ x ₁ ^∗ + p ₂ ⁰ x ₂ ^∗ > p ⁰ ₁ ω ₁ + p ₂ ⁰ ω ₂ p 1 z ₁ ⁰ + p 2 z ₂ ⁰ > 0 p ₁ ⁰ z 1 + p ⁰ ₂ z 2 > 0

Denition

A z ( p ) over-demand function satises a weak axiom of revealed

preferences (WARP), if by any p ⁰ 6= κ p price system p ⁰ z ( p ) ≥ 0

holds.

(49)

Uniqueness (cont.)

What are the conditions for the equilibrium to be unique? (It is not indierent whether there is one or more price systems!)

Statement

If in an economy the WARP holds for the z ( p ) over-demand

function, then the competitive equilibrium is unique.

(50)

Stability

How does the economic system behave if it is not in an equilibrium? (Dynamic analysis)

Price adaptation rule:

If D ( p ) − S ( p ) > 0 (over-demand), then p increases.

If D ( p ) − S ( p ) > 0 (over-supply), then p decreases.

Denition

The continuous and dynamic (Samuelson) price adaptation of a competitive economy:

p ˙ ( t ) = dp ( t )

dt = µ [ D ( p ( t ) − S ( p ( t )))] = (µ Z ( p ( t )))

With p 0 equilibrium price: p ˙ ( t ) = 0, thus D ( p 0 ) = S ( p 0 ) .

(51)

Stability (cont.)

Linear demand and supply, continuous price adaptation

D ( p ) ˙ = A − Bp , S ( p ) = C + Dp ( A , B , C , D > 0 ) p ˙ ( t ) = A − C

| {z }

α

+ (− B − D )

| {z }

β

p

p ˙ ( t ) = α + β p ( t ) (β < 0 )

Solution of the linear dierence equation: p ( t ) = e ^β ^t + c ₀ . The

equilibrium is stable if β < 0.

(52)

Stability (cont.)

Denition

Discrete and dynamic (Ezekiel) price adaptation rule of a competitive economy:

D ( p t )

Supply adapts to demand using the minus one period price:

S ( p t − 1 )

In equilibrium: D ( p _t ) = S ( p _t − 1 ) .

(53)

Stability (cont.)

Linear demand and supply, discrete price adaptation (Cobb-web model)

D ( p t ) ˙ = A − Bp t , S ( p t − 1 ) ˙ = C + Dp t − 1 ( A , B , C , D > 0 ) In equilibrium the demanded and supplied quantities equal.

D ( p t ) = A − Bp t = C + Dp t − 1 = S ( p t − 1 )

p t = A − C B

| {z }

α

+

− D B

| {z }

β

p t − 1

Linear dierence equation:

p _t = α + β p _t − 1

(54)

Stability (cont.)

Linear demand and supply, discrete price adaptation (Cobb-web model)

Linear dierence equation:

p t = α + β p t − 1

The p t = p t − 1 equilibrium is stable if |β| < 1, so that D < B

(supply is less responsive to price changes, than demand), prices

converge to the equilibrium.

(55)

Stability (cont.)

(56)

Stability (cont.)

The Samuelson-type continuous and dynamic price adaptation in a general equilibrium model:

p ˙ _i ( t ) = dp i ( t )

dt = µ _i [ D _i ( p ₁ ( t ), . . . , p _n ( t )) − S _i ( p ₁ ( t ), . . . , p _n ( t ))]

( i = 1 , . . . , n )

Statement

If the weak axiom of revealed preferences holds, then general

equilibrium is stable with the Samuelson-type continuous price

adaptation rule.

(57)

Market tests

'Tests' of the competitive market model.

Sellers and buyers are randomly chosen

Information: 'You have 100

tokens, which you can use to buy

a maximum of 3 pieces from a

good, with a price of 6. You can

sell the rst piece to the test

leader, for 16 tokens, the second

for 11 tokens, and the third for 3

tokens. When you repay the 100

tokens to the test leader the

remaining tokens is your prot,

which you can exchange to

dollars.'

(58)

Imperfect markets

Transaction costs: contracting, that is the cost of exchange.

Reasons:

Information asymmetry Property rights problems

Spatial distance bw. seller and buyer.

Etc.

(59)

Imperfect markets

Proportional transaction costs: We have to pay G fee after each

pieces of goods X .

(60)

Imperfect markets (cont.)

Due to the proportionate transaction costs selling any buying price will depart. The larger the distance, the more likely individuals will opt for self-supply, and the smaller the total volume of the

exchange.

(61)

Imperfect markets (cont.)

Lump-sum transaction costs: Lump-sum transaction costs do not create

disparities between selling and buying price. But they make consumers

to exchange only in discrete periods of time. So both buyers and sellers

are forced to keep endowments. Higher transaction costs and costs of

keeping endowment will raise the probability of self-supply and thus

decrease the volume of total exchange. In extreme cases the market

exchange nill.

(62)

Role of money

Money as the instrument of exchange

Money as tool for keeping value (transitionally)

(63)

Role of money (cont.)

Money decreases the costs of exchange. (It does not decrease the costs of physically exchanging goods. These exists in every economy with division of labor.) If only one good serves as the numeraire, less two way transaction is necessary. Moreover such a numeraire enables a three-, or more-way exchange, which is nearly impossible with barter. For exchange endowment of goods might be needed. Costs of exchange is the lowest of there is a consensus about the numeraire, used as money, and which can also serve as tool for keeping value.

Note

To analyze money, we need to include time in our analysis.

MICROECONOMICS II.