MICROECONOMICS II.

MICROECONOMICS II.

Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics,

Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department of Economics, Eötvös Loránd University Budapest

Institute of Economics, Hungarian Academy of Sciences Balassi Kiadó, Budapest

Author: Gergely K®hegyi Supervised by Gergely K®hegyi

February 2011

ELTE Faculty of Social Sciences, Department of Economics

MICROECONOMICS II.

week 2

General equilibrium theory, part 1

Gergely K®hegyi

Prepared by: Gergely K®hegyi, using Jack Hirshleifer, Amihai Glazer és David Hirshleifer (2009) Mikro- ökonómia. Budapest: Osiris Kiadó, ELTECON-könyvek (henceforth: HGH), and Kertesi Gábor (ed.) (2004) Mikroökonómia el®adásvázlatok. http://econ.core.hu/ kertesi/kertesimikro/ (henceforth: KG).

Introduction

Partial equilibrium

So far we have analyzed the partial equilibrium.

General equilibrium

General equilibrium

1. 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.

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

Perfect competition 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) 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) One participant, two goods, no production

Endowment: ω₁, ω₂

No market, so no exchange

Consumer optimum ('general eq.'): x^∗₁=ω₁, x^∗₂=ω₂

Robinson Crusoe economy

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.: <sup>∂U</sup><sub>∂c</sub> >0,<sup>∂U</sup><sub>∂`</sub> >0)

• Production function: c=f(h)(cond.: f⁰ >0, f⁰⁰<0) Robinson's decision:

• Maximize: U(c, `)→max<sub>c,`</sub>

• 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 = ^∂U_∂h +λ_dh^df = 0

−∂U/∂h

∂U/∂c = df dh

−M Uh

M U_c =M RSh,c=mph

∂U

∂h <0,∂U

∂c >0,df dh >0

−M Uh

M U_c =M RSh,c=mph

∂U

∂h <0,∂U

∂c >0,df dh >0

Robinson Crusoe economy

1. 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.

2. Assumption. Price-taker Robinson considers given:

• price of coconut: p

• wage: w

1. 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.

Robinson as producer

• Maximize: π=pcS−whD→maxc_S,h_D

• Subject to: c_S =f(h_D)

• Lagrange-function: L=pcS−whD−λT(cS−f(hD))

• FOC:

_∂c^∂L_S =p−λT = 0 _∂h^∂L_D =−w+λT df

dh_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)

Familiar optimum condition

pmph=w mph= w

p

Robinson as consumer

• Maximize: U(C_D, `)→max<sub>cD,`</sub>

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

• Maximize: U(CD, hS)→maxc_D,h_S

• Subject to: pcD=whS+π^∗ (whS: wage as labor)

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

• FOC:

_∂c^∂L_D =_∂c^∂U

D −λFp= 0 _∂h^∂L_S =_∂h^∂U

S +λ_Fw= 0

• Optimum condition:

−M U_c M Uh

=M RS_c,h= p w

• Solution:

coconut demand function: c_D(p, w) labor supply function: h_S(p, w)

leisure time demand function: `(p, w) = ¯`−h_S(p, w) Familiar optimum condition

−M Uc

M Uh

=M RSc,h= p w

Equilibrium in Robinson Crusoe economy

• Product (coconut) market: cD(p, w) =cS(p, w)

• Factor (labor) market: hD(p, w) =hS(p, w)

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

2. 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. letw=1˙ .

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

Returns to scale problems in a Robinson Crusoe economy

Problems of convexity in a Robinson Crusoe economy Without convexity equilibrium might not exist.

Robinson Crusoe economy Example:

• Robinson's utility function: U(c, `) =c<sup>2</sup>`

• Robinson's production function: f(h) =√ h

Pure exchange

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?

Edgeworth-Box

Edgeworth-Box

Set of allocations preferred by both parties

Pareto-ecient allocation

Set of Pareto-ecient allocations

Contract curve

Final set of allocations with a given set of endowments

Establishing the Pareto-ecient set of allocations Role of the social planner:

• Maximize:

U_A(x^A₁, x^A₂)→ max

x^A₁,x^A₂,x^B₁,x^B₂

• With subject to:

UB(x^B₁, x^B₂) = ¯UB

x^A₁ +x^B₁ =ω^A₁ +ω^B₁ x^A₂ +x^B₂ =ω^A₂ +ω^B₂

• Lagrange-function:

L=U_A(x^A₁, x^A₂)−λ U_B(x^B₁, x^B₂)−U¯_B

− µ1 x^A₁ +x^B₁ −ω₁^A−ω^B₁

−µ2 x^A₂ +x^B₂ −ω^A₂ −ω^B₂

• FOC:

1. _∂x^∂LA 1

= ^∂U_∂xA^A

1

−µ₁= 0 2. _∂x^∂LA

2

= ^∂U_∂xA^A 2

−µ2= 0 3. _∂x^∂LB

1

=−λ^∂U_∂xB^B 1

−µ₁= 0 4. _∂x^∂LB

2

=−λ^∂U_∂xB^B 2

−µ2= 0

M RSA= µ1

µ2

M RS_B =µ₁ µ2

• Contract curve (as an implicit function):

M RS_A(x^A₁, x^A₂) =M RS_B(x^A₁, x^A₂)

3. Note. Using x^A₁ +x^B₁ =ω₁^A+ω^B₁ andx^A₂ +x^B₂ =ω₂^A+ω₂^B border conditions the contract curve can be written as (e.g.) x^A₂ =ϕ(x^A₁).

Decentralized decisions The consumer

• max.:

UA(x^A₁, x^A₂)→ max

x^A₁,x^A₂

• to.: p₁x^A₁ +p₂x^A₂ =p₁ω₁^A+p₂ω₂^A

• opt. cond: M RSA=−^p_p¹

2

• demand func.: x^A₁(p1, p2), x^A₂(p1, p2) B consumer

• max.:

UB(x^B₁, x^B₂)→ max

x^B₁,x^B₂

• to.: p1x^B₁ +p2x^B₂ =p1ω₁^B+p2ω₂^B

• opt. cond.: M RSB =−^p_p¹

2

• demand func.: x^B₁(p1, p2), x^B₂(p1, p2)

M RSA=−p1

p2

, M RSB=−p1

p2

Market equilibrium

At equilibrium prices (p^∗₁, p^∗₂), demand equals supply on each market.

x^A₁(p^∗₁, p^∗₂) +x^B₁(p^∗₁, p^∗₂)

| {z }

D₁(p^∗₁,p^∗₂)

=ω₁^A+ω^B₁

| {z }

S₁(p^∗₁,p^∗₂)

x^A₂(p^∗₁, p^∗₂) +x^B₂(p^∗₁, p^∗₂)

| {z }

D2(p^∗₁,p^∗₂)

=ω₂^A+ω^B₂

| {z }

S2(p^∗₁,p^∗₂)

Establishing general equilibrium Two-good, two-party, pure exchange:

• parameters: p1, p2, x^A₁, x^A₂, x^B₁, x^B₂ (2 prices+2*2 consumed quantities=6 )

• number of equations (2+2+2=6):

optimum conditions (M RS-conditions): 2 (2 participant, 2 goods) budget constraints: 2 (two participant)

optimum conditions: 2 (two markets)

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

4. 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).

1. 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,

wherez1(p1, p2) =x^A₁(p1, p2)−ω₁^A+x^B₁(p1, p2)−ω₁^B andz2(p1, p2) =x^A₂(p1, p2)−ω^A₂ +x^B₂(p1, p2)−ω^B₂. 1. Proof. Let's summarize the budget constraints of the two consumer and rearrange it:

p1x^A₁ +p2x^A₂ ≡p1ω₁^A+p2ω₂^A p1x^B₁ +p2x^B₂ ≡p1ω₁^B+p2ω^B₂

p₁x^A₁ −p₁ω^A₁ +p₁x^B₁ −p₁ω^B₁

| {z }

p₁z₁(p₁,p₂)

+p₂x^A₂ −p₂ω^A₂ +p₂x^B₂ −p₂ω^B₂

| {z }

p₂z₂(p₁,p₂)

p1z1(p1, p2) +p2z2(p1, p2)≡0

3. 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.

4. 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).

Transaction (net) demand (supply) 2. Denition. Transaction (net)

• demand: x^t_i(p1, p2) ˙=xi(p1, p2)−ωi>0

• supply: x^t_i(p1, p2) ˙=xi(p1, p2)−ωi<0

Total and transaction individual demand (supply)

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.

Algorithm of nding equilibrium on a pure exchange economy 2. 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

Example:

• U_A=x^A₁x^A₂, ω^A₁ = 80, ω^A₂ = 30

• U_B=x^B₁x^B₂, ω₁^B= 20, ω₂^B= 70 Solution:

• Contract curve: x^A₂ =x^A₁

• Competitive equilibrium: x^A₁ = 55, x^A₂ = 55, x^B₁ = 45, x^B₂ = 45

5. Note. The above algorithm can be applied in anN product,M party pure exchange economy.

Finding the general equilibrium with M participant andN 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).

• 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.

6. 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).

Welfare theorems in a pure exchange economy

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

2. Proof. In a two-good, two-party, pure economy we saw that in a market equilibrium M RS_A = −^p_p^∗1∗ 2

and M RSB =−^p_p^∗1∗

2, since individual decisions are optimal. Thus M RSA =M RSB, 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 theM-party, N-product economy. .

3. 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 en- dowment of goods which leads the market participants to the above allocation of goods through decentralized decisions (market mechanism) (provided some technical conditions hold).

3. Proof. In a two-good, two-party, pure economy, whereω^A₁, ω₂^A, ω₁^B, ω₂^B are random endowments of goods, letx¯^A₁,x¯^A₂,x¯^B₁,x¯^B₂ be a Pareto-ecent allocation.

• Then M RS_A(¯x^A₁,x¯^A₂) =M RS_B(¯x^B₁,x¯^B₂) ˙=^p_p⁰¹0

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

• If p⁰₁x¯^A₁ + ¯x^A₂ =p⁰₁ω^A₁ +ω^A₂ (and then due to the condition above p⁰₁x¯^B₁ + ¯x^B₂ =p⁰₁ω^B₁ +ω₂^B), so this allocation is on the exchange line derived from (p⁰₁, p⁰₂), then the competitive mechanism concludes directly in the x¯^A₁,¯x^A₂,x¯^B₁,x¯^B₂ allocation as the competitive equilibrium.

• If p⁰₁x¯^A₁ + ¯x^A₂ 6=p⁰₁ω₁^A+ω^A₂, so the allocation was not on the exchange line derived from (p⁰₁, p⁰₂)then let us redistribute the original endowments:

∆ω₁^A=ω˙ ₁^A−ω¯₁^A,∆ω^A₂=ω˙ ^A₂ −ω¯^A₂,

∆ω₁^B=ω˙ ₁^B−ω¯₁^B,∆ω^B₂=ω˙ ₂^B−ω¯^B₂, (¯ω^A₁ + ¯ω^B₁ =ω^A₁ +ω^B₁,ω¯₂^A+ ¯ω₂^B=ω₂^A+ω₂^B)

so that p⁰₁x¯^A₁ + ¯x^A₂ = p⁰₁ω¯^A₁ + ¯ω^A₂. Then, after redistribution, the competitive mechanism concludes directly in the x¯^A₁,¯x^A₂,x¯^B₁,x¯^B₂ allocation as the competitive equilibrium.