MICROECONOMICS II.

Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041Course 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 SciencesBalassi Kiadó, Budapest

Author: Gergely K®hegyi

Supervised by Gergely K®hegyi

February 2011

1

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 equilibriumSo far we have analyzed the partial equilibrium.

General equilibrium

2

General equilibrium

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

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

Perfect competitionPerfect 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 e�ects)

• Market participants (consumers and sellers) are price-takers

• Total pro�t is allocated to consumers

• Everyone is rational (consumers maximize utility and sellers maximize pro�t)

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 exchangeConsumer optimum ('general eq.'): x∗1 = ω1, x

∗2 = ω2

3

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.: ∂U∂c > 0, ∂U∂` > 0)

• Production function: c = f(h) (cond.: f ′ > 0, f ′′ < 0)

Robinson's decision:

• Maximize: U(c, `)→ maxc,`

• 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 dfdh = 0

−∂U/∂h∂U/∂c

=df

dh

−MUh

MUc= MRSh,c = mph

∂U

∂h< 0,

∂U

∂c> 0,

df

dh> 0

−MUh

MUc= MRSh,c = mph

∂U

∂h< 0,

∂U

∂c> 0,

df

dh> 0

4

Robinson Crusoe economy

1. Assumption. A 'schizophrenic' Robinson: his producer and consumer side separates. Makes his decisionas price taker both on the demand and on the supply side, and then "meets" himself both on the factor andon 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 (pro�t willbe 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 → maxcS ,hD

• Subject to: cS = f(hD)

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

• FOC:

� ∂L∂cS

= p− λT = 0

� ∂L∂hD

= −w + λTdf

dhD= 0

5

• Optimum condition:pmph = w

mph =w

p

• Solution:

� coconut supply function: cS(p, w)

� labor demand function: hD(p, w)

� pro�t function: π(p, w)

Familiar optimum conditionpmph = w

mph =w

p

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: pcD = whS + π∗ (whS : wage as labor)

• Lagrange-function: L = U(CD, hS)− λF (pcD − whS − π∗)

• FOC:

� ∂L∂cD

= ∂U∂cD− λF p = 0

� ∂L∂hS

= ∂U∂hS

+ λFw = 0

6

• Optimum condition:

−MUc

MUh= MRSc,h =

p

w

• Solution:

� coconut demand function: cD(p, w)

� labor supply function: hS(p, w)

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

Familiar optimum condition

−MUc

MUh= MRSc,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 NOMONEY ILLUSION), one of the products or factors can be the numeraire. E.g. let w=1.

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

7

Returns to scale problems in a Robinson Crusoe economy

8

Problems of convexity in a Robinson Crusoe economyWithout convexity equilibrium might not exist.

Robinson Crusoe economyExample:

• Robinson's utility function: U(c, `) = c2`

• 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 bene�cial?

• When is it bene�cial?

9

Edgeworth-Box

Edgeworth-Box

Set of allocations preferred by both parties

10

Pareto-e�cient allocation

Set of Pareto-e�cient allocations

Contract curve

Final set of allocations with a given set of endowments

11

Establishing the Pareto-e�cient set of allocationsRole of the social planner:

• Maximize:UA(xA1 , x

A2 )→ max

xA1 ,xA

2 ,xB1 ,xB

2

• With subject to:

� UB(xB1 , xB2 ) = UB

� xA1 + xB1 = ωA1 + ωB

1

� xA2 + xB2 = ωA2 + ωB

2

• Lagrange-function:L = UA(xA1 , x

A2 )− λ

(UB(xB1 , x

B2 )− UB

)−

µ1

(xA1 + xB1 − ωA

1 − ωB1

)− µ2

(xA2 + xB2 − ωA

2 − ωB2

)• FOC:

1. ∂L∂xA

1= ∂UA

∂xA1− µ1 = 0

2. ∂L∂xA

2= ∂UA

∂xA2− µ2 = 0

3. ∂L∂xB

1= −λ∂UB

∂xB1− µ1 = 0

4. ∂L∂xB

2= −λ∂UB

∂xB2− µ2 = 0

MRSA =µ1

µ2

MRSB =µ1

µ2

• Contract curve (as an implicit function):

MRSA(xA1 , xA2 ) = MRSB(xA1 , x

A2 )

3. Note. Using xA1 + xB1 = ωA1 + ωB

1 and xA2 + xB2 = ωA2 + ωB

2 border conditions the contract curve can bewritten as (e.g.) xA2 = ϕ(xA1 ).

Decentralized decisions

The consumer

• max.:UA(xA1 , x

A2 )→ max

xA1 ,xA

2

• to.: p1xA1 + p2x

A2 = p1ω

A1 + p2ω

A2

• opt. cond: MRSA = −p1

p2

• demand func.: xA1 (p1, p2), xA2 (p1, p2)

B consumer

• max.:UB(xB1 , x

B2 )→ max

xB1 ,xB

2

12

• to.: p1xB1 + p2x

B2 = p1ω

B1 + p2ω

B2

• opt. cond.: MRSB = −p1

p2

• demand func.: xB1 (p1, p2), xB2 (p1, p2)

MRSA = −p1p2, MRSB = −p1

p2

Market equilibriumAt equilibrium prices (p∗1, p

∗2), demand equals supply on each market.

xA1 (p∗1, p∗2) + xB1 (p∗1, p

∗2)︸ ︷︷ ︸

D1(p∗1 ,p

∗2)

= ωA1 + ωB

1︸ ︷︷ ︸S1(p∗

1 ,p∗2)

xA2 (p∗1, p∗2) + xB2 (p∗1, p

∗2)︸ ︷︷ ︸

D2(p∗1 ,p

∗2)

= ωA2 + ωB

2︸ ︷︷ ︸S2(p∗

1 ,p∗2)

Establishing general equilibriumTwo-good, two-party, pure exchange:

• parameters: p1, p2, xA1 , x

A2 , x

B1 , x

B2 (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)

13

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 thegoods can be the numeraire. E.g. let p2=1. So the system of equations seems over determined (moreequations than parameter).

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

p1z1(p1, p2) + p2z2(p1, p2) ≡ 0,

where z1(p1, p2) = xA1 (p1, p2)−ωA1 +xB1 (p1, p2)−ωB

1 and z2(p1, p2) = xA2 (p1, p2)−ωA2 +xB2 (p1, p2)−ωB

2 .

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

p1xA1 + p2x

A2 ≡ p1ωA

1 + p2ωA2

p1xB1 + p2x

B2 ≡ p1ωB

1 + p2ωB2

p1xA1 − p1ωA

1 + p1xB1 − p1ωB

1︸ ︷︷ ︸p1z1(p1,p2)

+ p2xA2 − p2ωA

2 + p2xB2 − p2ωB

2︸ ︷︷ ︸p2z2(p1,p2)

p1z1(p1, p2) + p2z2(p1, p2) ≡ 0

3. Consequence. Due to the Walras-law equilibrium conditions cannot be independent (two equilibriumequations, one parameter (price)). So it is su�cient to use only one of the equations and the system will notbe over determined.

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

Transaction (net) demand (supply)

2. De�nition. Transaction (net)

• demand: xti(p1, p2)=xi(p1, p2)− ωi > 0

• supply: xti(p1, p2)=xi(p1, p2)− ωi < 0

14

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 (rede�ning 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

15

Example:

• UA = xA1 xA2 , ω

A1 = 80, ωA

2 = 30

• UB = xB1 xB2 , ω

B1 = 20, ωB

2 = 70

Solution:

• Contract curve: xA2 = xA1

• Competitive equilibrium: xA1 = 55, xA2 = 55, xB1 = 45, xB2 = 45

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

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 theLagrange 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 canbe 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 withthe algorithm.

6. Note. Counting the number of equations does not necessarily lead to a good conclusion, because negativeprices 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 existenceof equilibrium (see below).

Welfare theorems in a pure exchange economy

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

2. Proof. In a two-good, two-party, pure economy we saw that in a market equilibrium MRSA = −p∗1

p∗2

and MRSB = −p∗1

p∗2, since individual decisions are optimal. Thus MRSA = MRSB, so in equilibrium the

consumed basket of goods are on the contract curve, thus belong to the Pareto-e�cient allocation. This resultcan be generalized to the M -party, N -product economy. .

16

3. Statement. 2nd fundamental theorem of welfare economics: If the preferences of the market participantsare convex, then we can �nd a price system to any Pareto-e�cient allocation � with appropriately chosen en-dowment of goods � which leads the market participants to the above allocation of goods through decentralizeddecisions (market mechanism) (provided some technical conditions hold).

3. Proof. In a two-good, two-party, pure economy, where ωA1 , ω

A2 , ω

B1 , ω

B2 are random endowments of goods,

let xA1 , xA2 , x

B1 , x

B2 be a Pareto-e�cent allocation.

• Then MRSA(xA1 , xA2 ) = MRSB(xB1 , x

B2 )=

p01

p02can be the equilibrium price. With the numeraire being

p02=1.

• If p01xA1 + xA2 = p01ω

A1 + ωA

2 (and then due to the condition above p01xB1 + xB2 = p01ω

B1 + ωB

2 ), so thisallocation is on the exchange line derived from (p01, p

02), then the competitive mechanism concludes

directly in the xA1 , xA2 , x

B1 , x

B2 allocation as the competitive equilibrium.

• If p01xA1 + xA2 6= p01ω

A1 + ωA

2 , so the allocation was not on the exchange line derived from (p01, p02) then

let us redistribute the original endowments:

∆ωA1 =ωA

1 − ωA1 ,∆ω

A2 =ωA

2 − ωA2 ,

∆ωB1 =ωB

1 − ωB1 ,∆ω

B2 =ωB

2 − ωB2 ,

(ωA1 + ωB

1 = ωA1 + ωB

1 , ωA2 + ωB

2 = ωA2 + ωB

2 )

so that p01xA1 + xA2 = p01ω

A1 + ωA

2 . Then, after redistribution, the competitive mechanism concludesdirectly in the xA1 , x

A2 , x

B1 , x

B2 allocation as the competitive equilibrium.

17