MICROECONOMICS II.

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MICROECONOMICS II.

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ELTE Faculty of Social Sciences, Department of Economics

Microeconomics II.

week 2

GENERAL EQUILIBRIUM THEORY, PART 1 Author: Gergely K®hegyi

Supervised by Gergely K®hegyi

February 2011

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week 2 Gergely K®hegyi

Introduction Robinson Crusoe economy Pure exchange

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 Introduction

2 Robinson Crusoe economy

3 Pure exchange

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Partial equilibrium

So far we have analyzed the partial equilibrium.

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General equilibrium

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General equilibrium

Denition

The distribution of goods and the level of prices are called general equilibrium, if all demand and supply and factor demand and supply stem from individual optimization, and if all aggregate demand and aggregate supply are equal on each market.

Note

Non-perfect competition markets can have general equilibrium as well.

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Perfect competition

The good is homogeneous, divisible, and private No time (no money)

No insecurity

Perfect, immediate information (only price mediates info) Only market exchanges (no external eects)

Market participants (consumers and sellers) are price-takers Total prot is allocated to consumers

Everyone is rational (consumers maximize utility and sellers maximize prot)

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Possible models

One participant, one good, no production (uninteresting) One participant, more goods, no production (uninteresting) One participant, one production factor, one good (Robinson Crusoe economy)

One participant, one production factor, more goods One participant, more production factors, one good One participant, more production factors, more goods More participants, more goods, no production (only exchange)

More participants, more goods, one production factor More participants, one good, more production factors More participants, more goods, more production factors (exchange of produced goods)

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One participant, two goods, no production

Endowment: ω₁, ω₂ No market, so no exchange

Consumer optimum ('general eq.'):

x₁^∗=ω₁,x₂^∗=ω₂

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Robinson Crusoe economy

1 participant (Robinson), 1 good (coconut), 1 factor of production (labor)

Coconut consumption (pc): c

Leisure time 'consumption' (hours) : `(note: 0≤`≤`¯, e.g.:

`¯=˙24)

Working time (hours): h (note.: h= ¯`−`) Utility function: U(c, `)(cond.: ^∂_∂^U_c >0,^∂_∂`^U >0) Production function: c =f(h)(cond.: f⁰>0,f⁰⁰<0)

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Robinson Crusoe economy (cont.)

Robinson's decision:

Maximize: U(c, `)→max_c,`

Subject to:

c=f(h) h= ¯`−` Lagrange-function:

L=U(c,`¯−h)−λ(c−f(h)) First order condition:

∂L

∂c =^∂_∂^U_c −λ=0

∂L

∂h =^∂_∂h^U+λ^df_dh =0

−∂U/∂h

∂U/∂c = df dh

−MUh

MUc =MRSh,c =mph

∂U

∂h <0,∂U

∂c >0,df dh >0

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Robinson Crusoe economy (cont.)

−MU_h

MUc =MRSh,c =mph

∂U

∂h <0,∂U

∂c >0,df dh >0

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Robinson Crusoe economy

Assumption

A 'schizophrenic' Robinson: his producer and consumer side separates. Makes his decision as price taker both on the demand and on the supply side, and then "meets" himself both on the factor and on the product market for exchange.

Assumption

Price-taker Robinson considers given:

price of coconut: p wage: w

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Robinson Crusoe economy (cont.)

Algorithm

We solve the optimization exercises for both the producer and the consumer (prot will be paid to the owner).

We establish the demand and the supply, and the factor demand and supply curves.

We note the product and factor market equilibrium conditions.

We establish the equilibrium (product and factor) prices.

Using the demand and supply curves we establish the equilibrium quantities.

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Robinson as producer

Maximize: π=pcS−whD →maxcS,hD

Subject to: c_S =f(h_D)

Lagrange-function: L=pc_S −wh_D−λ_T(c_S −f(h_D)) FOC:

∂L

∂cS =p−λ_T =0

∂h∂L_D =−w+λ_T_dh^df

D =0 Optimum condition:

pmph=w mp_h= w

p Solution:

coconut supply function: cS(p,w) labor demand function: hD(p,w) prot function: π(p,w)

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Robinson as producer (cont.)

Familiar optimum condition

pmph=w mph= w

p

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Robinson as consumer

Maximize: U(CD, `)→maxcD,`

Subject to: pcD+w`=w`¯+π^∗ (π^∗: capital income as owner)

or

Maximize: U(CD,hS)→maxcD,hS

Subject to: pc_D =wh_S+π^∗ (wh_S: wage as labor)

Lagrange-function: L=U(C_D,h_S)−λ_F(pc_D −wh_S −π^∗) FOC:

∂L

∂cD =_∂^∂_c^U

D −λ_Fp=0

∂L

∂h_S =_∂^∂_h^U

S +λ_Fw=0 Optimum condition:

−MUc

MU_h =MRS_c,h= p w

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Robinson as consumer (cont.)

Solution:

coconut demand function: cD(p,w) labor supply function: hS(p,w)

leisure time demand function: `(p,w) = ¯`−hS(p,w)

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Robinson as consumer (cont.)

Familiar optimum condition

−MU_c

MU_h =MRS_c,h= p w

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Equilibrium in Robinson Crusoe economy

Product (coconut) market: c_D(p,w) =c_S(p,w) Factor (labor) market: h_D(p,w) =h_S(p,w)

Solution (general equilibrium): p^∗,w^∗,c^∗,h^∗, `^∗, π^∗,U^∗

Note

Since the (product and factor) demand and supply functions are zero order homogeneous (so NO MONEY ILLUSION), one of the products or factors can be the numeraire. E.g. let w=˙1.

Consequence

Equilibrium conditions cannot form independent system of equations (two equilibrium equations, one unknown price). So one equilibrium equation is enough.

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Equilibrium in Robinson Crusoe economy (cont.)

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Returns to scale problems in a Robinson Crusoe

economy

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Returns to scale problems in a Robinson Crusoe

economy (cont.)

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Problems of convexity in a Robinson Crusoe economy

Without convexity equilibrium might not exist.

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Robinson Crusoe economy

Example:

Robinson's utility function: U(c, `) =c²` Robinson's production function: f(h) =√

h Solution:

w^∗=1,p^∗=√

32,h^∗=12, `^∗=12,c^∗=√

12, π^∗=8

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Pure exchange

Two participant (A and B), two products (1 and 2), no production (no companies).

Consumers exchange their endowments.

Is exchange benecial?

When is it benecial?

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Edgeworth-Box

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Edgeworth-Box

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Set of allocations preferred by both parties

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Pareto-ecient allocation

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Set of Pareto-ecient allocations

Contract curve

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Final set of allocations with a given set of

endowments

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Establishing the Pareto-ecient set of allocations

Role of the social planner:

Maximize:

UA(x₁^A,x₂^A)→ max

x₁^A,x₂^A,x₁^B,x₂^B

With subject to:

UB(x₁^B,x₂^B) = ¯UB

x1Â+x1^B=ω₁Â+ω₁^B x₂Â+x₂^B=ω₂Â+ω₂^B Lagrange-function:

L=U_A(x₁^A,x₂^A)−λ U_B(x₁^B,x₂^B)−U¯_B

− µ₁ x₁^A+x₁^B−ω₁^A−ω^B₁

−µ₂ x₂^A+x₂^B−ω^A₂ −ω₂^B

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Establishing the Pareto-ecient set of allocations (cont.)

FOC:

1 ∂L

∂x₁^A =^∂U_∂x_A^A

1 −µ₁=0

2 ∂L

∂x₂^A =^∂U_∂x_A^A

2 −µ₂=0

3 ∂L

∂x₁^B =−λ^∂U^B

∂x^B₁ −µ₁=0

4 ∂L

∂x₂^B =−λ^∂^U^B

∂x^B₂ −µ₂=0

MRSA=µ₁ µ₂ MRS_B= µ₁ µ₂ Contract curve (as an implicit function):

MRSA(x₁Â,x₂Â) =MRSB(x₁Â,x₂Â)

Note

Using x₁Â+x₁^B =ωÂ₁ +ω₁^B and x₂Â+x₂^B=ωÂ₂ +ω₂^B border conditions the contract curve can be written as (e.g.) x₂Â=ϕ(x₁Â).

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Decentralized decisions

The consumer

max.:

UA(x₁^A,x₂^A)→ max

x₁^A,x₂^A

to.:p₁x₁Â+p₂x₂Â=p₁ω₁Â+p₂ωÂ₂ opt. cond: MRSA=−^p_p¹

2

demand func.:

x₁^A(p1,p2),x₂^A(p1,p2)

B consumer

max.:

U_B(x₁^B,x₂^B)→ max

x₁^B,x₂^B

to.:p₁x₁^B+p₂x₂^B=p₁ω^B₁+p₂ω₂^B opt. cond.: MRSB =−^p_p¹

2

demand func.:

x₁^B(p₁,p₂),x₂^B(p₁,p₂)

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Decentralized decisions (cont.)

MRS_A=−p₁

p₂,MRS_B=−p₁ p₂

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Market equilibrium

At equilibrium prices(p₁^∗,p^∗₂), demand equals supply on each market.

x₁^A(p₁^∗,p₂^∗) +x₁^B(p^∗₁,p₂^∗)

| {z }

D1(p₁^∗,p₂^∗)

=ω^A₁ +ω₁^B

| {z }

S1(p^∗₁,p₂^∗)

x₂^A(p₁^∗,p₂^∗) +x₂^B(p^∗₁,p₂^∗)

| {z }

D2(p₁^∗,p₂^∗)

=ω^A₂ +ω₂^B

| {z }

S2(p^∗₁,p₂^∗)

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Market equilibrium (cont.)

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Establishing general equilibrium

Two-good, two-party, pure exchange:

parameters: p1,p2,x₁^A,x₂^A,x₁^B,x₂^B (2 prices+2*2 consumed quantities=6 )

number of equations (2+2+2=6):

optimum conditions (MRS-conditions): 2 (2 participant, 2 goods)

budget constraints: 2 (two participant) optimum conditions: 2 (two markets)

Consequence

Number of equations and the number of parameters to estimate is equal.

Note

Since demand functions are zero order homogeneous (so NO MONEY ILLUSION), one of the goods can be the numeraire. E.g.

let p₂=˙1. So the system of equations seems over determined (more equations than parameter).

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Establishing general equilibrium (cont.)

Statement

Walras-law: The total value of demanded and supplied goods equals on the markets, so aggregate over-demand is always (with every price) zero.

p1z1(p1,p2) +p2z2(p1,p2)≡0,

where z1(p1,p2) =x₁Â(p1,p2)−ω₁Â+x₁^B(p1,p2)−ω^B₁ and z2(p1,p2) =x₂Â(p1,p2)−ω₂Â+x₂^B(p1,p2)−ω₂^B.

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Establishing general equilibrium (cont.)

Proof

Let's summarize the budget constraints of the two consumer and rearrange it:

p₁x₁Â+p₂x₂Â≡p₁ωÂ₁ +p₂ω₂Â p₁x₁^B+p₂x₂^B≡p₁ω^B₁ +p₂ω₂^B

p₁x₁^A−p₁ω^A₁ +p₁x₁^B−p₁ω₁^B

| {z }

p1z1(p1,p2)

+p₂x₂^A−p₂ω^A₂ +p₂x₂^B−p₂ω^B₂

| {z }

p2z2(p1,p2)

p₁z₁(p₁,p₂) +p₂z₂(p₁,p₂)≡0

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Establishing general equilibrium (cont.)

Consequence

Due to the Walras-law equilibrium conditions cannot be

independent (two equilibrium equations, one parameter (price)).

So it is sucient to use only one of the equations and the system will not be over determined.

Consequence

Due to the Walras-law if n−1 clears (is in equilibrium), then the n^th will clear as well (will be in equilibrium).

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Transaction (net) demand (supply)

Denition

Transaction (net)

demand: x_i^t(p₁,p₂) ˙=x_i(p₁,p₂)−ω_i >0 supply: x_i^t(p1,p2) ˙=xi(p1,p2)−ω_i <0

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Total and transaction individual demand (supply)

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Total and transaction market demand (supply)

The intersection of the aggregate supply and aggregate demand curve gives the total quantity of consumed goods in an economy, which has to equal the total supply quantity. The intersection of the aggregate transaction supply and aggregate transaction demand sets the quantity which is, in fact, exchanged. The two intersection points are at the same price level.

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Algorithm of nding equilibrium on a pure exchange economy

Algorithm

Writing the individual (consumer) optimum equations Solving these (establishing the demand functions)

Writing the market equilibrium conditions (demand=supply on each market)

Setting the numeraire (redening the demand functions so that they depend on the price ratio)

Establishing the equilibrium price ratio (one equation can be dropped)

Establishing the consumed quantities

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Algorithm of nding equilibrium on a pure exchange economy (cont.)

Example:

UA=x₁Âx₂Â, ω₁Â=80, ω₂Â=30 UB =x₁^Bx₂^B, ω^B₁ =20, ω₂^B=70 Solution:

Contract curve: x₂^A=x₁^A

Competitive equilibrium: x₁^A=55,x₂^A=55,x₁^B=45,x₂^B=45

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Algorithm of nding equilibrium on a pure exchange economy (cont.)

Note

The above algorithm can be applied in an N product, M party pure exchange economy.

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Finding the general equilibrium with M participant and N products

Parameters:

M∗N (N pc. goods, M pc. participants) N pc. prices

Number of parameters: M∗N+N Equations:

M∗N pc. individual optimum condition (rst order conditions + budget constraints for the Lagrange variables) N pc. equilibrium condition: aggregate demand = aggregate supply (total endowments)

Number of equations: M∗N+N

So the number of equations and parameters are equal.

BUT, since only relative prices matter (demand functions are zero order homogeneous), numeraire can be choosen (-1 parameter).

So the system seems over determined (more equation than parameters).

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Finding the general equilibrium with M participant and N products (cont.)

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

So the system is not over determined. Dropping one equation the equilibrium can be determined with the algorithm.

Note

Counting the number of equations does not necessarily lead to a good conclusion, because negative prices can also turn out. The reason is that budget constraints are in fact inequalities rather than equalities, and equilibrium conditions are also inequalities rather than equalities!→This is the problem of the existence of equilibrium (see below).

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Welfare theorems in a pure exchange economy

Statement

1st fundamental theorem of welfare economics: Competitive equlibrium is a Pareto-ecient state (provided some technical conditions hold).

Proof

In a two-good, two-party, pure economy we saw that in a market equilibrium MRSA=−^p_p¹^∗∗

2 and MRSB =−^p_p¹^∗∗

2, since individual decisions are optimal. Thus MRS_A=MRS_B, so in equilibrium the consumed basket of goods are on the contract curve, thus belong to the Pareto-ecient allocation. This result can be generalized to the M-party, N-product economy. .

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Welfare theorems in a pure exchange economy (cont.)

Statement

2nd fundamental theorem of welfare economics: If the preferences of the market participants are convex, then we can nd a price system to any Pareto-ecient allocation with appropriately chosen endowment of goods which leads the market participants to the above allocation of goods through decentralized decisions (market mechanism) (provided some technical conditions hold).

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Welfare theorems in a pure exchange economy

(cont.)

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Welfare theorems in a pure exchange economy (cont.)

Proof

In a two-good, two-party, pure economy, whereω₁Â, ω₂Â, ω^B₁, ω^B₂ are random endowments of goods, letx¯₁Â,¯x₂Â,x¯₁^B,x¯₂^B be a

Pareto-ecent allocation.

Then MRS_A(¯x₁^A,¯x₂^A) =MRS_B(¯x₁^B,¯x₂^B) ˙=^p_p⁰¹₀

2 can be the equilibrium price. With the numeraire being p⁰₂=˙1.

If p⁰₁¯x₁Â+ ¯x₂Â=p₁⁰ω₁Â+ω₂Â(and then due to the condition above p₁⁰x¯₁^B+ ¯x₂^B =p₁⁰ω^B₁ +ω₂^B), so this allocation is on the exchange line derived from(p₁⁰,p⁰₂), then the competitive mechanism concludes directly in thex¯₁Â,¯x₂Â,x¯₁^B,x¯₂^B allocation as the competitive equilibrium.

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Welfare theorems in a pure exchange economy (cont.)

Proof

If p⁰₁¯x₁Â+ ¯x₂Â6=p₁⁰ω₁Â+ω₂Â, so the allocation was not on the exchange line derived from(p₁⁰,p⁰₂)then let us redistribute the original endowments:

∆ωÂ₁=ω˙ Â₁ −ω¯₁Â,∆ωÂ₂=ω˙ ₂Â−ω¯₂Â,

∆ω₁^B=ω˙ ^B₁ −ω¯₁^B,∆ω^B₂=ω˙ ₂^B−ω¯^B₂,

(¯ω₁Â+ ¯ω^B₁ =ωÂ₁ +ω₁^B,ω¯₂Â+ ¯ω^B₂ =ωÂ₂ +ω₂^B) so that p⁰₁¯x₁Â+ ¯x₂Â=p₁⁰ω¯Â₁ + ¯ω₂Â. Then, after redistribution, the competitive mechanism concludes directly in the

x¯₁^A,¯x₂^A,¯x₁^B,¯x₂^B allocation as the competitive equilibrium.