Download Seminar 1 General Equilibrium and Welfare

Seminar 1
General Equilibrium and Welfare
Simona Montagnana
Week 21
19 February 2017
Introduction
Exchange economy model
Who I am
Simona Montagnana
[email protected]
http://people.bath.ac.uk/sm2446/
Office: 4.27 3East
Office Hours
Mondays from 13:00 to 15:00
Tuesdays from 13:00 to 15:00
Room: 2.16 3East
Welfare Economics
References
Introduction
Exchange economy model
Welfare Economics
References
Questions
2. Write down a pure exchange economy model and answer the
following:
a. What is the core of the economy? Explain the difference
between the concepts of contract curve and core of the
economy.
b. The Walrasian equilibrium must be in the core of the
economy. True or false.
Justify your answer.
3. Using an exchange economy general equilibrium model,
explain carefully the implications of the two fundamental
theorems of welfare economics for the role of the market and
the social planner in a welfarist society.
Introduction
Exchange economy model
Welfare Economics
References
Assumptions:
I
In the pure exchange economy model the only kind of
economic agent is the consumer→ economic activity:
consumption and trading (no production).
I
Each consumer i is described completely by her or his
preference (or utility function, ui ), and her or his initial
endowment of the k commodities wi .
I
Each consumer is assumed to behave competitively: the
prices as given, independent of her or his actions.
I
Each consumer attempts to choose the most preferred bundle
that she or he can afford.
Introduction
Exchange economy model
Welfare Economics
For simplicity:
I
Two goods 1 and 2,
I
Economic agent in the system: two consumers A and B,
I
A set of allocation (consumption bundle) for A and B:
(x1A , x2A ), (x1B , x2B )
I
Two utility functions (preferences) for A and B:
u A (x1A , x2A ), u B (x1B , x2B )
I
Consumer’s preferences are strongly monotone.
References
Introduction
Exchange economy model
Welfare Economics
References
I
Two endowments for consumer A and B: w A (w1A , w2A ), w B (w1B , w2B )
I
Total endowment:
W = W A + W B = (w1A + w1B , w2A + w2B ) = (w1 , w2 )
I
A feasible allocation is any distribution of good 1 and 2 among the
consumers A and B (any combination is possible):
(x1A , x2A ), (x1B , x2B ), s.t.
x1A + x1B = w1A + w1B = w1
x2A + x2B = w2A + w2B = w2
I
The initial endowment is a particular feasible allocation that
consumers start with, and consists of the amount of each good that
the consumers bring to the market.
I
The consumers trade the goods among themselves according to
certain rules (they are price-takers). They will trade and end up
with a final allocation.
Introduction
Exchange economy model
Welfare Economics
References
Core & Contract curve
a. What is the core of the economy? Explain the difference
between the concepts of contract curve and core of the
economy.
Def. ”Core of an economy”: A feasible allocation x is in the
core of the economy if it cannot be improved upon by any coalition
(group of individuals acting on their own using only the total
endowment of the group itself).
In the case of two-consumers and two-goods the core is the subset
of the Pareto set at which each agent does better than by refusing
to trade. In other words, the core is simply that segment of the
Pareto set that lies between the indifference curves that pass
through the initial endowment W .
Def. ”Contract curve”: the set of all Pareto efficient allocations.
Introduction
Exchange economy model
Welfare Economics
References
Every allocation in the core is also in the contract curve,
but not viceversa.
Graphically:
XB1
0B
Pareto Set: C,D,E, all Pareto efficient points
XB2
E
XA2
Contract
curve
D
C
Core of
Contract
Curve
W
0A
Endowmentpoint
XA1
W1
W2
Introduction
Exchange economy model
Welfare Economics
References
Differences:
Remember that a Pareto efficient allocation is one for which
there is no way to make all agents better off. Said another way, a
Pareto efficient allocation is one for which each agent is as well off
as possible, given the utilities of the other agents (Varian).
Pareto optimality concept :
I
is only concerned with efficiency and has nothing to say about
distribution of welfare.
I
depends only of the total endowment and not on the
endowment point.
I
consumers’ preferences (IC).
Core of an economy:
I
depends only of the endowment point.
I
consumers’ preferences (IC).
Introduction
Exchange economy model
Welfare Economics
References
Walrasian equilibrium
b. The Walrasian equilibrium must be in the core of the
economy. True or false. Justify your answer.
True.
An allocation (x i )i∈N is in the core ifPthere isP
no group of
individuals S and (x̂ i )i∈N such that
x̂ji =
wji for each good
i∈S
i∈S
j, and U i (x̂ i ) > U i (x i ) for each individual i ∈ S.
Introduction
Exchange economy model
Welfare Economics
References
Proof: Assume that the Walrasian equilibrium x ∗i is not in the
core. Then there is:
S ⊂ I ⇒ (x̂ i )i∈S , s.t:
P i
P i
i.
x̂j =
wj ∀j
i∈S
i∈S
ii. U i (x̂ i ) > U i (x ∗i ) for each individual i ∈ S.
But the definition of the Walrasian equilibrium implies:
PP
PP
pj x̂ji >
pj wji
i∈S j
⇒
P
j
pj
i∈S j
X
i∈S
P
x̂ji
>
P
j
pj
P
i∈S
wji
| {z }
i∈S
wji
which contradicts the first equality (i).
Economic intuition: the value of the alternative allocation strictly
preferred by this coalition must be larger than their endowment,
which means this allocation is not affordable for them.
Introduction
Exchange economy model
Welfare Economics
References
3. Using an exchange economy general equilibrium model,
explain carefully the implications of the two fundamental
theorems of welfare economics for the role of the market and
the social planner in a welfarist society.
Def. ”Welfare Economics or Normative Economics”: branch
of economics that is concerned with the appropriate allocation of
resources within an economy and with the establishment of
criteria or norms that can evaluate or judge the policies that are
likely to maximise social welfare.
In the words of Baumol ”Welfare Economics has concerned itself
mostly with policy issues which arise out of the allocation of
resources, with the distribution of inputs among the various
commodities and the distribution of commodities among various
consumers” .
Introduction
Exchange economy model
Welfare Economics
References
The First Fundamental Theorem of Welfare Economics says if
preferences are stricktly monotonic, and if (x1∗ , x2∗ , p) is an
equilibrium outcome, then the allocation (x1∗ , x2∗ ) is Pareto optimal.
⇒ every Walrasian equilibrium is a Pareto optimum.
Conditions:
I
competitive market, that means the consumers are price taker ⇒
but the idea of competitive equilibrium make sense when there are
enough agents. In the Edgeworth box there are only two agents,
that can recognize their market power and use it to improve their
position;
I
no externalities, each person’s utility depends, only on his or her
own bundle of goods,
I
no asymmetric information.
Introduction
Exchange economy model
Welfare Economics
References
Implications FTWE:
I
a private market that is competitive will result in Pareto efficiency all gains from trade will be exhausted;
I
a competitive market is a benchmark by which policy-makers can
judge actual market outcomes;
I
the theorem assumes that there are no market imperfections such as
monopoly, externalities and public goods;
I
agents need to know the prices of goods to make their consumption
decisions.
Introduction
Exchange economy model
Welfare Economics
References
The Second Fundamental Theorem of Welfare Economics says
if the preferences are convex, continuous, and monotonic, a Pareto
efficient allocation is a Walrasian equilibrium for some set of prices.
⇒ every Pareto optimum allocation can be decetralised as a
Walrasian equilibrium.
Conditions:
I
The preferences have to be convex, continuous, and monotonic.
Implications:
I
The issue of efficiency and the issue of equality distribution are two
distinct issues.
I
The market mechanism is distributionally neutral.
I
The price plays two different roles in the market system: an
allocative and a distributive role.
I
The distributional issues should be resolved by changing the initial
endowments (lump-sum transfer), and not by changing the
competitive pricing mechanism.
Introduction
Exchange economy model
Welfare Economics
The idea of a Welfare Function is to consider all together the
different consumers’ utilities, and construct some kind of social
preferences:
Individual welfare refers to the utility associated to a single
consumer or economic agent.
I Social welfare is the utility associated to society as a whole.
I
U"lity
PossibilitySet
U"lity
Possibility
Fron"er
Social
Op"mality
References
Introduction
Exchange economy model
Welfare Economics
U"lity
PossibilitySet
Asocialplanner
willselectan
alloca"ononthe
contractcurve,
Eachpointonthe
contractcurveis
associatedwitha
pair(U1;U2).
U"lity
Possibility
Fron"er
Thesetofpoints
generatesthe
U0lityPossibility
Fron0er,
Pointsaandbare
Pareto-efficient,
butpointcis
inefficien.
Social
Op"mality
Thealloca"onis
chosenthat
achievesthe
highestlevelof
socialwelfare,
Pointoonthe
u"litypossibility
locusachievesthe
highestsocial
indifferentcurve.
References
Introduction
Exchange economy model
Welfare Economics
References
Social Welfare Function
The form of the social welfare function (SWF) will determine
the shape of the social indifference curves. The SWF captures the
ethical objectives of the society.
Along the social indifferent curve a social planner is indifferent
among different points on the curve.
I
How do we represent, from a mathematical point of view, this
trade off across people?
the SWF is an aggregating function, that has the same features of
the utility functions:
W (ui , ..., un )
The social welfare function can be construct differently,
depending on:
I
I
social norms,
policy maker or social planner’s aim.
Introduction
Exchange economy model
Welfare Economics
References
Examples of Social Welfare Functions
Pure utilitarian or Benthamin WF: a Social Welfare Function
that care of the total amount of individual utility functions.
With two consumers A and B:
UB
W (uA , uB ) = uA + uB
In general:
W (ui , ..., un ) =
n
X
i=1
ui
UA
Utilities of each member are given equal weight.
The isowelfare lines have slope of −1, indicating that the utilities of
both are treated equally at the margin. Loosely speaking we can say
that one extra unit of utility for a starving person is not seen to be
of any greater value than an extra unit of utility for a millionaire.
I Fair? Not quite.
I
I
Introduction
Exchange economy model
Welfare Economics
References
A more generalized function as this form:
With two consumers A and B:
UB
W (uA , uB ) = αuA + βuB
In general:
W (ui , ..., un ) =
n
X
i=1
I
αi ui
UA
Where different members of society ought to be given different
weight ⇒ αn > 0 reflect the importance of the members of society
(example: give greater weight to adults, hardworking people, etc.).
Introduction
Exchange economy model
Welfare Economics
References
Minimax or Rawlsian WF: a Social Welfare Function that cares
of the allocation of welfare of the worst off agent.
With two consumers A and B:
UB
W (uA , uB ) = Min(uA , uB )
In general:
W (ui , ..., un ) = Min(ui , ...un )
UA
I
This SW describes an equity seeking behaviour in the distribution of
utility.
I
In this case the social planner will try to maximise the welfare of the
least well-off person in the society: choosing allocations that
maximizes the minimum utility.
I
Improvements in the utilities of the richest do not improve the SW.
Introduction
Exchange economy model
Welfare Economics
Social optimum
To achieve a social optimum, we have to face with a
maximization of society’s welfare, that is defined by two
concepts:
I
efficiency: Concerned with the optimal production and allocation
of resources given existing factors of production.
I
equity: Concerned with how resources are distributed throughout
society.
References
Introduction
Exchange economy model
Welfare Economics
References
An allocation of resources is (socially) optimal, if it satisfies the
following two criteria:
1. It must be an efficient allocation.
I
Efficiency - synonymous with Pareto optimality.
I
A resource allocation is Pareto optimal if it is impossible to make
one member of society better off without making some other
member or members worse off.
2. It must be an equitable allocation.
I
Equity is a normative concept, with no universal definition (what
is equitable to you, not necessarily is equitable to me).
Introduction
Exchange economy model
Welfare Economics
References
Readings
Varian (2014), Intermediate Microeconomics: A Modern Approach
(Ninth Edition), Ch. 32 and 34.