General Equilibrium and Welfare Exercise 1

Exercise 1

General Equilibrium and WelfareExercise 1

Simona Montagnana

Dep. Economics, University of Bath

March 4, 2019

30/06/18, 00:59Word Art 1

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Simona Montagnana Question 2-3 1

Exercise 1

Question 2

Write down a pure exchange economy model with two individuals andtwo goods. State explicitly the key assumptions of your model. Using yourmodel, explain carefully the implications of the two fundamental theoremsof welfare economics for the role of the market and the social planner in awelfarist society.


Exercise 1

Assumptions

Economic agent: consumers.

Economic activity: trading & consumption.

Each individual is described completely by:

own utility function,own initial endowment of commodities.No externalities or public goods.

Individuals’ preferences are monotonic.

Perfect and complete markets.

Competitive behaviour.

Rationality.

Complete information.


Exercise 1

Model

For simplicity:

Two goods: 1 & 2,

Two consumers: A & B,

A set of allocation (consumption bundle): (xA1 , xA2 ), (xB1 , x

B2 )

Two utility functions (preferences): uA(xA1 , xA2 ), uB(xB1 , x

B2 )

Two endowments for consumer A & B:wA = (wA

1 ,wA2 ),wB = (wB

1 ,wB2 )

Total endowment in the economy:W = W A + W B = (wA

1 + wB1 ,w

A2 + wB

2 ) = (w1,w2)

A feasible allocation is any combination of good 1 & 2 among consumer A& B: (xA1 , x

A2 ), (xB1 , x

B2 ), s.t.

xA1 + xB1 = wA1 + wB

1 = w1

xA2 + xB2 = wA2 + wB

2 = w2


Exercise 1

FTWE

The First Fundamental Theorem of Welfare Economics says ifpreferences are stricktly monotonic, and if (x1; x2; p) is an equilibriumoutcome, then the allocation (x1; x2) is Pareto optimal.

Every Walrasian equilibrium is Pareto Efficient.

Conditions:

Monotonic preferences.

Competitive behaviour

Rationality

No market imperfections such as externalities and public goods.

Complete information.

Limits:

the location of the competitive equilibrium allocation is highlydependent on the location of initial endowment.

Not necessary a ”social optimal” allocation.


Exercise 1

STWE

The Second Fundamental Theorem of Welfare Economics says if thepreferences are convex, continuous, and monotonic, a Pareto efficientallocation is a Walrasian equilibrium for some set of prices.

Every Pareto efficient allocation can be decetralised as a Walrasianequilibrium.

The theorem stating that any given Pareto-efficient allocation in aperfectly competitive market is a general equilibrium outcome for someinitial allocation.

Conditions:The preferences have to be convex, continuous, and monotonic.

All the FTWE’s conditions.

Solution:Lump-sum transfers: transfer to or from an individual for which the size isunaffected by the individual’s choices.


Exercise 1

Market ⇔ Social Planner - Graphically

wA2

wA1

wB1

wB2

0A

0B

1

2

1

2

E*

-P1*/P2*

E’

W’

-P1’/P2’

Laissez-faireTarget

Figure: Efficiency and social optimality in a two-person economySimona Montagnana Question 2-3 7

Exercise 1

Implications of FTWE

A market that is competitive will result a Pareto efficient outcome -all gains from trade will be exhausted.

A competitive market is a benchmark by which policy-makers canjudge actual market outcomes.


Exercise 1

Implications of STWE

The market mechanism is distributionally neutral.

The price plays two different roles in the market system:

an allocative role is to indicate relative scarcity.a distributive role is to determine how much of different goods eachconsumer can buy.

The issue of efficiency and the issue of equality distribution are twodistinct issues.

The government can play a positive role.

The distributional issues should be resolved by changing the initialendowments (lump-sum transfer), and not by changing thecompetitive pricing mechanism.


Exercise 1

Role of the Market & Social Planner

The role of the market is to achieve an efficient outcome throughthe prices’ mechanism.

The role of the social planner is to achieve an social optimumthrough a distribution of welfare.


Exercise 1

Intuition on Market & Social Planner

The ”first theorem of welfare economics” shows that a Walrasianequilibrium is Pareto efficient, but the social welfare consequences ofthis result are limited because of the role played by initialendowments.

The location of the Walrasian equilibrium at E ∗ into diagram abovewas significantly influenced by the designation of W ∗ as the startingpoint for trading. Points on the contract curve outside the range ofconsumers A & B are not attainable through voluntary transactions,even though these may in fact be more socially desirable than E ∗.

The second theorem of welfare economics” addresses this issuethought ”transfers” (w ′w∗). It states that for any Pareto optimalallocation of resources there exists a set of initial endowments and arelated price vector such that this allocation E ′ is also a Walrasianequilibrium.


Exercise 1

Fairness & Efficiency


Exercise 1

Pareto Efficiency vs. Social Optimum

Pareto Efficiency is only concerned with the optimality of anallocation, but it has nothing to say about distribution of welfare orSocial Optimum.

There are many Pareto Efficient Allocations, but we may not knowwhich Pareto Efficient allocation with should have. To deal with thisissue, we have to introduce the social welfare function.


Exercise 1

Positive vs. Normative Economics

Positive Economics: Analysis of How Things Really Are (e.g., Dohigher taxes reduce labor supply? What are the costs and benefits ofimplementing a congestion charge?)

Normative Economics: Analysis of How Things Should be (e.g.,should the government intervene in health insurance market? Howhigh should taxes be?)


Exercise 1

Welfare Economics

Welfare economics focuses on using resources optimally to achieve themaximum well-being for the individuals in society. Welfare economicsconcerned with what is good and what is bad” rather than what is.

”The main problem of welfare economics deals with may be formulated asfollowed: What conditions have to be fulfilled by a community of individualin order to make its welfare a maximum?” (Jan Tinbergen, May 1957).

Government intervention

In most cases the government will be faced with a variety of policyoptions, with benefits and costs of each policy varying for individualsacross society.


Exercise 1

Towards Social choice

We know how individuals measure the preference, but how should societyvalue them?This is the fundamental question of social choice.

Problem:

How can we aggregate individual preferences into social or grouppreferences?


Exercise 1

Social Welfare

A social welfare function (SWF) measures the welfare of societyas a function of utilities of all individuals.

In other words, SWF is a function that aggregates individualutilities, u(x):

W = F (u1(x), u2(x), ..., un(x))

The objectives were to establish a complete social ordering of allpossible alternative states of the world and to determine the sociallyoptimal choice across policy options.


Exercise 1

Social choice and Welfare

The SWF captures the ethical objectives of the society.Along the social indifferent curve a social planner is indifferent amongdifferent points on the curve. The social welfare function can be constructdifferently, depending on:

Social norms,

Policy maker or social planner’s aim.

In this way, an SWF formalizes the philosophy, the morality and thenorms behind the choices made by society, allowing us to clearly identifythe moral choices we make, and to consider their consequences.

The government will need to choose among different SWF, and will betaking a moral decision in making that choice.


Exercise 1

From Contract Curve to UPF

The Utility Possibilities Frontier shows the levels of utility such thateach of two individuals achieve when they have traded to an efficientoutcome on the contract curve.

0A

0B

1

2

1

2

E*E’

R

N

O

M

Figure: Contract Curve

UB

UA

E’

E*

U"lityPossibilityFron"erUPF

S

R

M

O

N

Figure: Utility Possibilities Frontier


Exercise 1

Equity and Efficiency

Each point on the contract curve is associated with a pair (UA;UB),st:

Points N,R,E ′,E∗,O correspond to point on the contract curve andare efficient.

At point S both individual would be better off, but is not feasible.

Point M is inefficient because individuals may trade (within blueshaded area - Pareto Improvement) and make both better off.

Suppose M and R are the only possible allocations. Is R moreequitable than M? Not necessary. Compared with M, R yields moreequity for consumer B and less for consumer A. Some people may feelthat M is more equitable than R; others may feel the opposite.


Exercise 1

Improvement towards Pareto Optimality

UB

UA

E’

E*

U"lityPossibilityFron"erUPF

UB*

UA*UA’

UB’

S

R

M

O

N

ParetoImprovement

Figure: Pareto Improvement

One inefficient allocation of resources may be more equitable than anotherefficient allocation.


Exercise 1

Social Welfare Maximisation

To achieve a social optimum, we have to face with a maximization ofsocietys welfare (SWF):

maxxA1 ,xA2 ,xB1 ,xB2W[UA(xA1 , x

A2 ),UB(xB1 , x

B2 )]

st:.

xA1 + xB1 = w1

xA2 + xB2 = w2


Exercise 1

Pareto Optimality & Social Optimum

Given the SWF, the allocation E ∗is chosen on the utility possibilityfrontier such that achieves the highest level of isowelfare curve:

An allocation of resources issocially optimal, if it satisfiesthe following two conditions:

1. It must be an efficientallocation.

2. It must be an equitableallocation.

UB

UA

E’

E*UB*

UA*UA’

UB’

R

O

N

IsoWelfarecurves

SocialOp:mum

M

Figure: Social Optimum


Exercise 1

Conditions of Social optimum

To achieve a social optimum, we have to face with a maximization ofsociety’s welfare, that is defined by two conditions:

efficiency: Concerned with the optimal allocation of resources givenexisting factors/resources.

equity: Concerned with how resources are distributed throughoutsociety.

Example:

Depending on what social welfare function we seek to maximize, societymight prefer an inefficient allocation to an efficient one. If allocation a givesBill Gates 1, 000, 000, 000 and 9 get nothing, it is Pareto efficient. Butsurely society would prefer allocation b where all 10 people get 100, 000Pareto efficient equity


Exercise 1

From UPF to Social Optimum

Introduction Exchange economy model Welfare Economics References

The idea of a Welfare Function is to consider all together thedi↵erent consumers’ utilities, and construct some kind of socialpreferences:

I Individual welfare refers to the utility associated to a singleconsumer or economic agent.

I Social welfare is the utility associated to society as a whole.

U"lityPossibilitySet

U"lityPossibilityFron"er

SocialOp"mality


Exercise 1

Moral Issue - Social Welfare Functions

The problem is how to define an equitable allocation?

Which social welfare function should we use?

The answer depends on what one thinks equity entails.

1 Bentham suggested that we should maximize the sum of peoplesutilities:

W = UA + UB

2 We could generalize the Benthamin SWF and give different peopledifferent weights:

W = αAUA + βBU

B with 0 ≤ α, β,and α + β = 1

3 Rawls suggested that we should seek to maximize the utility of theleast well or in society:

W = min(UA,UB)


Exercise 1

Graphically

The WSFs (1) and (2) take a particular (and extreme) stance oninequality aversion:

1 Implies that societys level of inequality aversion approaches infinity,2 Suggests that society does not care about inequality at all.

UB

UA

U%litarianorBenthaminSWF WeightedSWF RawlsianSWF

UB UB

UAUA

Figure: Social Welfare Functions


Exercise 1

SummaryIntroduction Exchange economy model Welfare Economics References

U"lityPossibilitySet

Asocialplannerwillselectan

alloca"ononthecontractcurve,

Eachpointonthecontractcurveisassociatedwithapair(U1;U2).

U"lityPossibilityFron"er

Thesetofpointsgeneratesthe

U0lityPossibilityFron0er,

PointsaandbarePareto-efficient,butpointcisinefficien.

SocialOp"mality

Thealloca"onischosenthatachievesthe

highestlevelofsocialwelfare,

Pointoontheu"litypossibilitylocusachievesthehighestsocial

indifferentcurve.


Exercise 1

Question 3

Write down a pure exchange economy model with n ≥ 2 individuals andm ≥ 2 goods. State explicitly the key assumptions of your model. Usingyour model, explain the following questions:

(a) It is possible to have a Pareto efficient allocation where everyone isworse off than they are at an allocation that is not Pareto efficient?True or false. Justify.

(b) If we know the contract curve, then we know the outcome of anytrading? True or false? Justify.

(c) Can some individual be made better off if we are at a Pareto efficientallocation? Justify.

(d) If the excess demands are equal to zero in m− 2 of the markets, whatmust be true about the remaining two markets? - true or false?Justify.


Exercise 1

Model

n consumers i = 1, 2...n

m commodities j = 1, 2...m

Consumer i ′s consumption set is Rm+ → (x ij )

Preference relation �i over consumption vector in this set → U i (x ij )

Each consumer i is initially endowed with an amount of w ij ≥ 0 of

good j .

The total endowment in the economy is denoted by w j =n∑

i=1w ij

A feasible allocation is any combination of m commodities among nconsumers:

α = {x |x ij ≥ 0 for all i , j andn∑

i=1x ij =

n∑i=1

w ij for all j}.

Assumption ⇒ see slide n.3.


Exercise 1

(a) It is possible to have a Pareto efficient allocation where everyone isworse off than they are at an allocation that is not Pareto efficient? Trueor false. Justify.

False, for this would mean that at the allegedly PE allocation there issome way to make everyone better off, contradicting the assumption of PE.

An outcome is a Pareto efficient allocation if it cannot be Paretoimproved. A Pareto efficient allocation is not Pareto dominate by anyother allocation.In other words, an efficient (or Pareto efficient) allocation of goods is anallocation in which no one can be made better off without makingsomeone else worse off.


Exercise 1

(b) If we know the contract curve, then we know the outcome of anytrading? True or false? Justify.

False, if we know the contract curve, then any trading should end upsomewhere in the curve; hover, we don’t know where.

Def. contract curve is the set of all Pareto efficient allocations.It shows all allocations from which no mutually beneficial trade con bemade.

Properties:

Once a point on a contract curve, such as S, has been chosen, thereis no way to move to another point on the contract curve, say R,without making one person worse off.


Exercise 1

(c) Can some individual be made better off if we are at a Pareto efficientallocation? Justify.

Yes, an individual can be better off, but not without making someone elseworse off the allocation will be Pareto inefficient.

An efficient allocation occurs when no one consumer can be made betteroff by trade without making someone else worse off.When consumers make all mutually advantageous trade, the outcome isefficient and lies on the contract curve, but that particular outcome maynot be equitable. Indeed, a consumer can be better off in an allocationwhich is inefficient.

Efficiency and equity are two separate concepts.


Exercise 1

(d) If the excess demands are equal to zero in m − 2 of the markets, whatmust be true about the remaining two markets?

In a model with n ≥ 2 individuals and m ≥ 2 goods, if the excess ofdemands (e) is zero, in m − 2, this means that:

m∑j=1

pjej = 0, ej = 0 for j = 1, ...,m − 2

Then must be that:

⇒ pm−1em−1 + pmem ≡ 0

This mean that the total value of aggregate excess of demands in theremaining two markets must be equal to zero.

In other words, we can state that the value of the excess demand in onemarket must equal to the value of the excess supply in the other.


Exercise 1

Extra notes on Walras’s Law

Definition

Z ij (p1, p2) = X i

j (p1, p2)− w ij

The value of aggregate excess demand is identically zero, for all set ofprices, not just in equilibrium.

Proof. In the case of two consumers and two goods:

p1xA1 (p1, p2) + p2x

A2 (p1, p2) ≡ p1w

A1 + p2w

A2

p1[xA1 (p1, p2)− wA1 ] + p2[xA2 (p1, p2)− wA

2 ] ≡ 0

p1eA1 (p1, p2) + p2e

A2 (p1, p2) ≡ 0 (1)

Same for B:...

p1eB1 (p1, p2) + p2e

B2 (p1, p2) ≡ 0 (2)


Exercise 1

Implication of Walras Law

Adding the last two equations (1) and (2), we get the Walras Law:

p1z1(p1, p2) + p2z2(p1, p2) ≡ 0 (3)

Each agent must satisfy his/her budget constraint for all prices.

Therefore if Z1(p∗1 , p∗2) = 0 then according to (3) also Z2(p∗1 , p

∗2) = 0 and

vice-versa.

An important implication of Walras Law is this: if (at some p >> 0),m − 1 markets clear, then the remaining market must also clear.

The key question is ’Why’?Because the equilibrium equations Z1(p∗1 , p

∗2) = 0 and Z2(p∗1 , p

∗2) = 0 are

not independent ⇒ due to the price normalisation.


Exercise 1

Questions & References

Any Question?

Please ask questions about the topic:using the office hours;sending emails;using the appropriate: Moodle or Padlet forum.

For Padlet: just click on the link, click on the message board, and askyour question anonymously. There is no need to create an account. Iwill check Padlet regularly, and answer questions so that everyone cansee them.

Just R.E., Hueth D.L., Schmitz A. (2004), The Welfare Economics ofPublic Policy, Edward Elgar ed.

Mas-Collel A., Whinston M.D., Green J.R. (1995), Microeconomictheory, Ch. 15.

Varian H.R (2014), Intermediate Microeconomics, ch. 32 and 34, 9ed.


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General Equilibrium and Welfare Exercise 1

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