Part IIa: Paper 1 General Equilibrium and Welfare Economics Dr Hamish Low

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Part IIa Paper 1General Equilibrium and
Welfare EconomicsDr Hamish Low

• Lecture 6

Supervision 3, question 2b need to understand
implications of Arrow for BS-SWF Office hour
11.45am-12.45 Monday 24th Nov
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Outline Efficiency and Equity

• Conditions for Pareto Efficiency
• First Fundamental Theorem
• Efficiency vs Equity and the Second Fundamental
  Theorem
• First Best vs Second Best
• Lump sum taxes

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Necessary Conditions for Pareto Efficiency

• First Best
• an allocation which is Pareto efficient subject
  only to overall resource constraints and
  technology constraint
• Second Best
• an allocation which is Pareto efficient subject
  to resource constraints and additional
  constraints (eg. imperfect information)

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First Best Pareto Efficiency

• Nothing about prices in defining first-best
  allocation
• All goods are private
• rivalrous (if one person consumes a good, this
  uses up the good and so no one else can consume
  it)
• excludable (one person can stop another from
  consuming the good)
• Conditions for Pareto Efficiency
• Production efficiency
• Consumption efficiency
• Overall efficiency

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Production Efficiency
subject to
F.O.C.s
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Production Efficiency
Oy
K
Area of Pareto improvement
Potential allocation of inputs
Ox
L
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Consumption Efficiency
OB
y
Area of Pareto improvement
Potential allocation of consumption goods
OA
x
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Overall Efficiency
y
Potential allocation for economy
Area of Pareto improvement
PPF
x
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Fundamental Theorems of Welfare Economics

• Introduce prices what are the welfare properties
  of the market?
• Assumption 1 Competitive Markets
• Consumers and firms take prices as given
• Consumers and firms optimise

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Assumption 2 Complete Markets

• Enforcability of contracts and complete
  allocation of property rights
• A market exists for all goods which influence
  either individual utilities or firm production

• No externalities
• No public goods
• No asymmetries of information

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(Aside on externalities)

• Consumption externality
• Assumption is NOT that this sort of utility
  function is ruled out
• Rather the assumption is that there is a price
  for each good

if there is a price (market) for each good,
then the externality becomes a pecuniary
externality
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First Fundamental Theorem

• If assumption of complete and competitive markets
  holds, then a general competitive equilibrium is
  first-best Pareto efficient
• Intuition
• Optimising consumers each set
• So, if all consumers face the same prices, then
  consumption efficiency
• Similarly with optimising firms each sets
• Allocation of production between firms

The invisible hand!
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Equity vs Efficiency

• FT1 market leads to the Pareto frontier
• Last week which point on the Pareto frontier
  maximises social welfare
• Is there a cost between moving to a point on the
  frontier which improves social welfare? ie a
  trade-off between efficiency and equity

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Assumption 3 Convexity

• Individual indifference curves are convex
• rules out a preference for extremes (over
  mixtures)
• Firm production sets are convex
• rules out increasing returns to scale
• (Assumptions necessary for existence of
  competitive equilibrium)

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Assumption 4 Ideal Lump-sum Taxes

• A lump sum tax or transfer is where the payment
  (or receipt) is independent of individual choices
• Transfer based on an individuals endowment
  (individual specific)
• Key imposition of a lump-sum tax leaves all
  individuals facing common prices
• but those common prices may depend on the lump
  sum taxes because of wealth effects

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Assumption 4 Ideal Lump-sum Taxes

• The poll tax introduced by Margaret Thatcher was
  not a lump-sum tax
• tax was paid per head
• could avoid the tax by not registering to vote
• amount of the tax did not vary with endowment
• A tax on labour income is not lump-sum
• the price paid by the firm is different from that
  received by the worker
• individuals can change their tax burden by
  changing their behaviour

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Second Fundamental Theorem

• If assumptions 1 to 3 hold, then any Pareto
  efficient allocation can be achieved by
  appropriate redistribution through an ILST and
  leaving the competitive market to find prices.

Prices will depend on endowments and the extent
of redistribution BUT all individuals face the
same prices so no distortion to Pareto efficiency
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FT2 in an Exchange Economy
Competitive equilibrium, no intervention
Competitive equilibrium that maximises SWF
OB
y
Initial endowment
SWF
e2
e1
OA
x

• To achieve SWF, tax is imposed on individual
  As endowment, and this is transferred to
  individual B
• Tax and transfer are lump-sum since value of
  endowment is taken as given by each individual

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(Aside on aggregation)

• When do reallocations in wealth have no effect on
  aggregate demand (and hence on prices)?
• Reallocation of income from person i to j means
  demand from j increases, i decreases. Exactly
  offset if both have equal income elasticities
• Implies income elasticity must be constant across
  individuals with different levels of income
• homothetic prefences
• necessary for representative agent models

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Implications of FT2

• Decentralisation result redistribute with
  limited government informational requirements
• decentralised decisions will lead to a first-best
  Pareto allocation (satisfy the marginal
  conditions)
• Separate out equity from efficiency any point on
  the Pareto frontier can be achieved.
• there is no conflict between redistribution and
  achieving a first-best Pareto allocation

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Implications

• There are two distinct reasons for government
  intervention (under the assumptions of the FT2)
• To redistribute competitive market may not lead
  to the Pareto efficient point that maximises
  social welfare
• To correct market failure (due to failure of one
  of assumptions 1 to 3)
• to return to first best (eg use of Pigouvian
  taxes to correct externalities)

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Value of FT2

• FT2 provides rationale for why government
  intervention may be desirable
• Assumptions (particularly on information) are
  very strong
• FT2 is not valuable to understand what form
  interventions should take
• second best analysis of policy intervention

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First Best vs Second Best

• Focus on FB vs SB for efficiency (ie keep
  assumption 4 that LST feasible)
• Distortion to FB there is one good where price
  does not equal marginal cost (eg price of using a
  road is zero)
• Competitive equilibrium will not be first best
• First best solution is it possible to remove the
  distortion directly? If so, then move back to the
  first best.
• If not
• try to satisfy as many Pareto conditions as
  possible (ie ensure efficiency in all other
  markets ignoring distortion )
• or introduce offsetting distortion (second best
  policy)

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Example

• Car use (when congestion)
• First best directly price externality (road
  pricing set at marginal cost imposed on others)
  Pigouvian solution
• Second best what is the price for public
  transport?
• Price at marginal cost public transport is at
  Pareto efficient level
• But cars and public transport are substitutes
  price below marginal cost induces switch away
  from car use
• 1 car user switches saving of deadweight loss in
  car market, p-mc
• imposes deadweight loss in public transport, but
  negligible since pmc

efficiency saving by introducing extra distortion
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CAR USE
PUBLIC TRANSPORT
Deadweight loss
p
p
SMC
Introduced deadweight loss from marginal increase
in q (P-MC)
Reduction in deadweight loss from marginal
reduction in q MB - SMC
Dd
Dd
MC
p
PMC
q
q
q
q
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Information and LST

• Redistribution requires lump sum payments to vary
  across individuals according to their
  characteristics (eg endowments of ability)
• These characteristics are private information to
  the individual. Government only observes
  behaviour.
• Needs to be in interests of the individuals to
  reveal information on characteristics through
  behaviour

Problem of incentive compatibility
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Lump Sum Taxes and Incentive Compatibility
1 individual of high ability, 1 individual of low
ability. Identical preferences over
consumption, c, and leisure, z Consider a LST
that depends on ability. Budget constraint for i
becomes Ti is the lump sum tax (which is a
transfer received if negative)
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• Government chooses Th and TL to maximise a
  Utilitarian SWF. Equivalently, government chooses
  consumption and leisure bundles for each person.

First-order conditions for consumption
Equate marginal utility of consumption across
individuals With identical utility functions,
this implies identical levels of consumption
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First-order conditions for leisure
Since, ah / al gt 1,
Same level of consumption, but high ability
person works more
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• If ability is unobservable, base tax on observed
  income
• First best requires that higher income
  individuals pay T and lower income individuals
  receive T, such that consumption equalises
• high ability lose incentive to work, instead earn
  low income to mimic low ability individuals

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Implications

• Impossible to differentiate lump-sum taxes by
  unobservable characteristics!
• Taxes must be based on observed behaviour (or
  characteristics)
• When characteristics are private information,
  redistribution must involve taxes that are
  distortionary.
• Policy prescriptions cannot be based on Second
  Welfare Theorem cannot separate out efficiency
  from equity

• Some conditions for first best Pareto Efficiency
  will not hold
• Theory of the second best suggests Pareto
  Efficiency in remaining markets may not be optimal

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Summary

• Necessary conditions for Pareto efficiency
• First Welfare Theorem competitive market leads
  to Pareto efficiency
• Second Welfare Theorem any Pareto efficient
  point is achievable by LST and competitive
  markets
• allows us to separate out efficiency from equity
• 2nd Best if distortion in one market, maybe
  optimal to introduce an extra distortion
  elsewhere to offset the first one (rather than
  trying to satisfy Pareto conditions)
• Unable to use lump sum taxes because of private
  information about endowments (of ability). Should
  not try to maintain separation between equity and
  efficiency