Lecture #3 - microeconomics.wne.uw.edu.pl€¦ · The pure exchange model: The case of 2 goods and 2 consumers −To simplify the analysis, we consider only 2 goods (X and Y) and

General EquilibriumPure Exchange

Lecture #3

• Partial equilibriumequality of demand and supply in a single market (an assumption: actions in one market do not influence, or have negligible influence on other markets)

• General equilibriumadjustment of demand and supply in all markets at the same time (including interactions between markets)

Example: Two interdependent markets and adjustment to the equilibrium

Goods are substitutes:− DVD movies (rental stores)− shows (cinema tickets)

The change in price in one market will influence the other.

Let− Pticket = 20 PLN− PDVD = 10 PLN− The government imposes a tax

of 2 PLN per cinema ticket.

http://pclab.pl/zdjecia/artykuly/pila/creative-s750_vs_logitech-z5500/filmyDVD.jpghttp://pclab.pl/zdjecia/artykuly/pila/creative-s750_vs_logitech-z5500/filmyDVD.jpg

DVDM

P

DVDs

P

Cinema tickets

SM

SV

20 PLN

Qk QV

10 PLN

21 PLN

Q’k

S*M

Tax => Decrease in supply

D’V

Q’V

12 PLN

Increase in the price for cinema tickets=> Increase of the demand for DVDs


P

DVDs

DM

SM

20 PLN

Qk

21 PLN

Q’k

S*M

Increase in the price of DVDs=> Increase of the demand for cinematickets and in the price of tickets

D’V

DV

SV

QV

10 PLN

Q’V

12 PLN

Influence of one market on the otherwill continue up to the point when the general equilibrium is reached

D*M

23 PLN

Q*kQ”k

22 PLN

D’M

13 PLN

Q*V

D*V

Cinema tickets

P


− With no analysis of interactions between the markets (as in the limited case of partial equilibrium analysis), the influence of the tax on the market for cinema tickets will be underestimated.

− For a similar situation but in the market for goods which are complements, this influence would be overestimated.


Marginal Utility, Marginal Rate of Substitution

Marginal utility (MU) – a change in the total utility resulting from the change in consumption of a given good by 1 unit.

𝑴𝑴𝑴𝑴𝑿𝑿 =∆𝑴𝑴∆𝑿𝑿

Marginal rate of substitution (MRS) – a measure indicating how many units of one good a consumer is willing to forego for an additional unit of another good, maintaining her utility at the same level.

𝑴𝑴𝑴𝑴𝑴𝑴𝒙𝒙𝒙𝒙 =∆𝒀𝒀∆𝑿𝑿

= −𝑴𝑴𝑴𝑴𝒙𝒙𝑴𝑴𝑴𝑴𝒙𝒙

MRS for different utility functions

Cobb-Douglas utility function – normal goods

U = Axayb => 𝑴𝑴𝑴𝑴𝑴𝑴𝒙𝒙𝒙𝒙 = −𝒂𝒂𝒃𝒃𝒙𝒙𝒙𝒙

Linear utility functions – substitutes

U = ax + by => 𝑴𝑴𝑴𝑴𝑴𝑴𝒙𝒙𝒙𝒙 = −𝒂𝒂𝒃𝒃

Leontief utility function – complementsU = min{ax,by} => no substitution between goods

Utility maximization

y

m/py

m/px x

y*

x*

For decreasing MRS (convex indifference curve):

𝐦𝐦𝐦𝐦𝐦𝐦 𝑴𝑴 ⇔ 𝑴𝑴𝑴𝑴𝑴𝑴𝒙𝒙𝒙𝒙 = −𝑷𝑷𝒙𝒙𝑷𝑷𝒙𝒙

MRSxy – slope of the indifference curve-Px/Py – slope of the budget line

Optimum (x*,y*) – a tangency point of the budget line and an indifference curve

The pure exchange model(only final goods, only consumers)

Each consumer has an endowment of goods at her disposal, and consumers can exchange these goods between themselves (a production process is omitted).

The pure exchange model:The case of 2 goods and 2 consumers− To simplify the analysis, we consider only

2 goods (X and Y) and 2 consumers (A and B).− This approach is more general than one could think:

o The second good may represent ”all other goods”.o Each consumer might be treated as a “consumer type”.

− Initial endowments: ωA = (ωXA, ωYA) and ωB = (ωXB, ωYB)

− For example, ωA = (7,1) means that consumer A has 7 units of good X and 1 unit of good Y.

− When ωA = (7,1) and ωB = (3,5), there is in total 7 + 3 units of good X in the market, and 1 + 5 units of good Y in the market.

Feasible allocations

− What are feasible allocations of 10 units of good X and of 6 units of good Y?

− A feasible allocation – the total amount of each good consumed is equal to the total amount available

− One of feasible allocations is the initial allocation of the endowments: ωA = (ωXA, ωYA) and ωB = (ωXB, ωYB)

− The Edgeworth box allows us to depict all feasible allocations of two goods between two consumers in one diagram

Edgeworth box

Box width = ωXA + ωXB – a number of units of X available in the market

Box height = ωYA + ωYB – a number of units of Y available in the market

10X Consumer B

Consumer A

6Y

10X6Y

YA

YB

XA

XB

1Y 5Y

3X

7X

P

Edgeworth box – Initial allocationωA = (ωXA, ωYA) = (7,1) and ωB = (ωXB, ωYB) = (3,5)

10X Consumer B

Consumer A

6Y

10X6Y

YA

YB

XA

XB

1Y 5Y

3X

7X

P

Edgeworth box – Initial allocationωA = (ωXA, ωYA) = (7,1) and ωB = (ωXB, ωYB) = (3,5)

P – Initial allocationAll allocations in the box (including the edges) are feasible.

What allocations are preferred by the two consumers?

Consumer A ωXA

ωYA

X

Y

Taking into account preferences

Consumer B ωXB

ωYB

X

Y


Consumer BωXB

ωYB

X

Y


UB = constUA = const

10X B

A

6Y

10X6Y

Benefits from exchange

P

In P: MRSA≠MRSBEach point within the grey area ismore beneficial (advantageous)

for both consumers

Edgeworth box

10X B

A

6Y

10X6YUB

1

UA1P UA2

UB2

R

Edgeworth box – Pure exchange

R gives higher utility to both consumers than P, but still there

is place for improvement

10X B

A

6Y

10X6YUB

1

UA1P UA2

UB2

RIn S, both MRS are equal– the allocation is Pareto efficient

UB3

S

Edgeworth box – Pure exchange

Pareto efficient allocation

• In a Pareto efficient allocation, it is not possible to improve the situation of any participant of the exchange without worsening the situation of the other one.

• In other words, a mutually advantageous exchange is not feasible.

• Within the Edgeworth box, there are numerous Pareto efficient allocations.

Consumer A

Consumer B

ωXA

ωYA

ωXB

ωYB

All points marked are Pareto efficient

Edgeworth box – Pareto efficient allocations

Edgeworth box – Contract curve

A

B

X

Y

The set of all Pareto efficient points is called the Pareto setor the contract curve.For each point on the contract curve MRSxyA = MRSxyB.

Final allocation

• Which allocation from the contract curve will be the market equilibrium?

• It depends on the bargaining power of the individuals in the market.

• Is it a perfect competition market?• Are there any rules of the trade?

Each consumer is a price-taker and maximizes her utility given the prices in the market and her initial endowment

A purely competitive market

Consumer A ωXAωYA

pXqXA + pYqYA = pXωXA + pYωYA

qYA*

qXA*

qYA

qXA


Consumer A ωXAωYA

pXqXA + pYqYA = pXωXA + pYωYA

qYA*

qXA*

• Initial endowment – amounts a consumer initially has: ωXA and ωYA

• Gross demand – amounts a consumer wants to consume: q*XA and q*YA

• Net demand – amounts a consumer wants to purchase: q*XA – ωXA and q*YA – ωYA

qYA

qXA


A

B

Budget line for Consumer A

ωXA

ωYA

qYA

qXAωYB

ωXB

qYB

qXB

would be optimal for A, but this is never optimal for B (not a tangency point).

A

B


ωXAωYA

qYA

qXAωYB

ωXB

qYB

qXB

• For prices pX and pY: – excess supply of good X – excess demand of good Y

• When total amounts of demanded goods X and Y are not equal to the available amounts of the goods, the market is in disequilibrium.

• The market prices needs to adjust.


Equilibrium in a purely competitive market

• In the equilibrium, prices are such that each consumer is choosing his or her most-preferred affordable bundle, and demand equals supply in every market.

• In the Edgeworth box, the equilibrium is in the tangency point of the indifference curves.

• The slope of the tangent line in this point is equal to the negative of the equilibrium price ratio - pX/pY.

• MRSA = MRSB = - pX/pY

A

B


ωXAωYA

qYA

qXAωYB

ωXB

qYB

qXB

• Excess supply of X – pX will decrease• Excess demand of Y – pY will increase• The slope of the budget line (-pX/pY) will decrease• The budget line will rotate around the initial endowment point

The example considered

Which allocations are possible equilibria in a competitive trade?

The example considered

A

B

ωXA

ωYA

qYA

qXA

ωYB

ωXB

qYB

qXB


• With the new prices, the markets for both goods (X and Y) are balanced, so we reach the general equilibrium.

• Trade in a competitive market yields a Pareto efficient allocation.

• This is known as the First Theorem of Welfare Economics.

Fundamental Theorems of Welfare Economics

andWalras’ Law

Market equilibrium||

Competitive equilibrium||

Walrasian equilibrium||

A set of prices such that each consumer is choosing her most-preferred affordable bundle, and demand

equals supply in every market

Leon Walras (1834-1910) – a French economist; an early investigator of general equilibrium theory


• For– a perfectly competitive market– with two consumers (A and B)– with convex indifference curves describing

preferences towards two goods (X and Y),the equilibrium in the Edgeworth box is in the tangency point of the indifference curves of the consumers.

• The slope of the tangent line in the equilibrium is equal to the negative of the equilibrium price ratio (–pX/pY).

• MRSA = MRSB = –pX/pY

General equilibrium

The total amounts of goods that consumers want to consume may differ from the available amounts of the goods.

The market is in disequilibrium.

Market prices need to adjust.

A change in one market causes a change in another market. This will be continued until the general equilibrium

(i.e., equilibrium in each market) is reached.

First Theorem of Welfare Economics

When • consumer preferences are well-behaved (locally

non-satiated) and• there exists a competitive equilibrium, then this (competitive) equilibrium is a Pareto efficient allocation.

Local non-satiation of preferences – for any original bundle of goods, there is another bundle of goods arbitrarily close to the original bundle, but that is preferred to the original one.

There are some implicit assumptions in this framework: e.g., all consumers care only about their own consumption, no externalities, no transaction costs, perfect information.

• It guarantees that a competitive market will exhaust all of the gains from trade: An equilibrium allocation achieved by a set of competitive markets will necessarily be Pareto efficient.

• However, such an allocation may not have other desirable properties.

• In particular, the theorem says nothing about the distribution of economic benefits.

• The market equilibrium might not be a “just” allocation: for example, if consumer A owned everything to begin with, then she would own everything after trade. That would be efficient, but it would probably not be fair.

First Theorem of Welfare EconomicsEfficiency versus Fairness

Second Theorem of Welfare Economics

• A converse to the First Welfare Theorem• It says that under some conditions, any Pareto efficient

allocation is (part of) a competitive equilibrium.

• When preferences are convex, there is always a set of prices such that each Pareto efficient allocation is a competitive equilibrium for some initial endowments of the goods.

• Each Pareto efficient allocation means each point from the contract curve.

• Again, there are some implicit assumptions in this framework: e.g., all consumers are utility-maximizers, no externalities, no transaction costs, perfect information.


Consumer A

Consumer B

Can the allocation in black be reached by competitive trading, starting from the initial endowment ω? No.


Consumer A

Consumer B

But the allocation in black can be reached by competitive trading, starting from the initial endowment θ.

In brief

• What is the relationship between competitive equilibrium and Pareto efficiency?

• First Welfare Theorem is about: Is any competitive equilibrium Pareto efficient?

• Second Welfare Theorem is about: Is any Pareto efficient allocation (part of) a competitive equilibrium?

Walras’ Law

• Net demand or excess demand – a difference between how much of some good a consumer wants to consume and how much of that good she initially has.

• The value of aggregate excess demand is zero for any positive prices.

• It is zero for all possible sets of prices, not just for the equilibrium prices.

• The value of aggregate excess demand – the values of excess demands summed over all markets in the economy.

Walras’ Law – Derivation

• Assumptions:– Two goods (1 and 2); two consumers (A and B)– Every consumer has well-behaved preferences– For any positive prices (p1,p2), each consumer spends all

of her budget

• Budget constraint of Consumer A

• Budget constraint of Consumer B

x – gross demand ω – initial endowment(x – ω) – net / excess

demand

We can sum them up

And rearrange


The budget constraints:

(x – ω) is net demand / excess demand.

Walras’ Law: The value of aggregate excess demand is zero for any positive prices.


Implication 1 of Walras’ Law

If the value of aggregate excess demand in k-1 markets is zero, then the value of excess demand in the remaining k-th market is also zero.

Suppose there is no excess demand for good 1:

Then, Walras’ Law

implies that


For an economy with two goods, excess supply in the market of one good implies excess demand in the market of the other good.

Suppose there is excess supply of goods 1:

Then, Walras’ Law

implies that


To determine the equilibrium for an economy consisting of k markets for different goods, we only need to find a set of equilibrium prices for k − 1 of these markets.

• Walras’ Law implies that the market for good k will automatically have demand equal to supply if all other markets in the economy are in equilibrium.

• This means there are really only k – 1 independent prices.• We can choose one of the k prices and set it equal to

a constant.• It is often convenient to set one of the prices equal to 1 so

that all other prices can be interpreted as being measured relative to it. Such a price is called a numeraire price.

Market Equilibrium and Planning

• It is difficult to reach the general equilibrium if all markets are not purely competitive.

• An efficient allocation may be achieved by central planning.

• However, market solutions are preferred (to central planning) because consumers and producers are able to better specify their preferences and production possibilities.

Slajd numer 1Slajd numer 2Example: Two interdependent markets and adjustment to the equilibriumSlajd numer 4Slajd numer 5Slajd numer 6Marginal Utility, �Marginal Rate of SubstitutionMRS for different utility functionsUtility maximization�The pure exchange model�(only final goods, only consumers)�The pure exchange model:�The case of 2 goods and 2 consumersFeasible allocationsEdgeworth boxSlajd numer 14Slajd numer 15Slajd numer 16Slajd numer 17Slajd numer 18Slajd numer 19Slajd numer 20Slajd numer 21Pareto efficient allocationSlajd numer 23Edgeworth box – Contract curveFinal allocationSlajd numer 26Slajd numer 27Slajd numer 28Slajd numer 29Slajd numer 30Slajd numer 31Slajd numer 32Slajd numer 33Slajd numer 34Market equilibrium�||�Competitive equilibrium�||�Walrasian equilibriumEquilibrium �in a purely competitive marketGeneral equilibriumFirst Theorem of Welfare EconomicsSlajd numer 39Second Theorem of Welfare EconomicsSecond Theorem of Welfare EconomicsSlajd numer 42In briefWalras’ LawSlajd numer 45Slajd numer 46Slajd numer 47Slajd numer 48Slajd numer 49Slajd numer 50Market Equilibrium and Planning

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Lecture #3 - microeconomics.wne.uw.edu.pl€¦ · The pure exchange model: The case of 2 goods and 2 consumers −To simplify the analysis, we consider only 2 goods (X and Y) and