Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
General equilibrium
Existence
• Counting equations and unknowns (Walras)
o n markets → n excess demand equations z j ( p)
but at the same time only n -1 unknown relative
prices
However if n-1 of the equations are satisfied the last
one will be satisfied as well (Walras’ law)
ƒ Walras’ law: ∑ p j z j = 0 . The total value of
excess demands is exactly zero
• Fixed Point Theorem (a Walrasian equilibrium where
agents are passive price takers)
o Define a mapping from a price vector p to a new
price vector by the following rules
ƒ If excess demand is positive add to the initial
price some multiple of the excess demand (increase price)
ƒ If excess demand is zero let the new price equal
the old price
ƒ If excess demand is negative add to the initial
price some multiple of the excess demand (decrease price), unless doing so would make the
new price negative, in which case set it instead
at zero
ƒ Normalize, making ∑ p j = 1
o Fixed Point Theorem: There exists a price vector p
that maps into itself, the equilibrium vector p*
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
• The shrinking core (an Edgeworth equilibrium where
agents are active market participants)
o Start with a small number of participants → the set
of possible outcomes is called the core. When the
number of participants increases the core shrinks,
until it in the limit only contains the competitive
market equilibrium (the Walrasian equilibrium)
(figure 13.2)
Stability and uniqueness
• Will not be dealt with in this course
Welfare economics
Definitions:
- Pareto superior
- Pareto efficient
- Utility frontier
Value judgements:
- Process independence
- Individualism
- Non-paternalism
- Benevolence
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Pareto efficiency conditions
A model with
two consumers, two inputs (h),
two firms, two goods (i)
A Pareto efficient allocation is the solution to the problem
u1 ( x11 , x12 , z1 )
max
s.t
s.t
u 2 ( x21 , x22 , z 2 ) = u 2
2
∑x
h =1
hi
≤ xi
i = 1, 2 (goods, consumptio n)
ih
≤ zh
h = 1, 2 (inputs)
2
∑z
i =1
xi = f i ( zi1 , zi 2 ) i = 1, 2 (goods, production )
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Consumers, inputs
Firms, goods
h = 1,2
i =1,2
Firm 1
z11
Firm 2
z21
z12
z22
z2
z1
x2
x1
x11
x12
x21
Consumer 1
x22
Consumer 2
x11 + x 21 ≤ x1
x12 + x 22 ≤ x 2
z11 + z 21 ≤ z1
z12 + z 22 ≤ z 2
u 1 ( x11 , x12 , z1 ) with marginal utilities u11 , u 12 and u 1z
u 2 ( x 21 , x 22 , z 2 ) with marginal utilities u12 , u 22 and u z2
x1 = f 1 ( z11 , z12 ) with marginal products f11 and f 21
x 2 = f 2 ( z 21 , z 22 ) with marginal products f 12 and f 22
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
The Lagrangean for the Pareto efficiency problem is
L = u1 ( x11 , x12 , z1 ) + λ [u 2 ( x21 , x22 , z 2 ) − u 2 ] +
⎡
⎤
⎡
⎤
+ ∑ ρ i ⎢ xi − ∑ xhi ⎥ + ∑ ω h ⎢ z h − ∑ zih ⎥ +
i
h
i
⎣
⎦ h
⎣
⎦
+ ∑ μ i f i ( zi1 , zi 2 ) − xi
[
]
i
First order conditions:
∂L / ∂x1i = ui1 − ρ i = 0
i = 1, 2
∂L / ∂x2 i = λ u − ρ i = 0
i = 1, 2
2
i
∂L / ∂z1 = u1z + ω1 = 0
efficient consumption
efficient input supply
∂L / ∂z 2 = λ u z2 + ω 2 = 0
∂L / ∂zih = μ i f hi − ω h = 0 i , h = 1, 2
efficient input use
∂L / ∂xi = ρ i − μ i = 0
efficient output mix
i = 1, 2
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Results:
Efficient consumption
See the Edgeworth box in figure 13.2
u11 ρ1 u12
MRS = 1 =
= 2 = MRS 212 with fixed labour supply and fixed outu2 ρ 2 u2
put
1
21
Efficient input supply (four conditions)
u zh ω h
MRS = − h =
= f hi
ρi
ui
h
iz
h, i = 1, 2
Marginal production of zh in the
production of commodity i
What consumer h requires in compensation for increasing his supply
of zh, expressed in units of commodity i
Efficient input use
MRTS
1
21
f11 ω1
f12
= 1=
= 2 = MRTS 212 See the Edgeworth box in figure
13.4 with fixed labour supply
f 2 ω2
f2
Efficient output mix
See figure 13.5 where
f12
f 22 ρ1 u11 u12
MRT21 = 1 = 1 =
= 1 = 2 = MRS 21 the conditions for efρ 2 u2 u2
f1
f2
ficient input use are
supposed to be met
Marginal rate of transformation Marginal rate of substitution bebetween the two goods
tween the two goods
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
The social welfare function
- There is more than one Pareto efficient allocation
- Pareto superiority does not yield a complete ordering of
allocations, in particular no ordering at all of Pareto efficient allocations
- A Pareto efficient allocation must not even be Pareto superior to a particular inefficient allocation
What is required is a social welfare preference ordering
which, just like the consumer’s preference ordering, provides
complete, transitive and reflexive comparisons of allocations
If the welfare preference ordering is continuous it can be represented by a function, often referred to as a
Bergsonian social welfare function W, (swf)
A Paretian swf is a Bergsonian swf embodying the value
judgements of:
(1) individualism
W = W ( x11 , x12 , z1 , x21 , x22 , z 2 )
(2) non-paternalism
(
)
W = W u1 ( x11 , x12 , z1 ), u 2 ( x21 , x22 , z2 ) = W (u1 , u 2 )
(3) benevolence
∂W (u 1 , u 2 )
= Wh > 0
h
∂u
h = 1,2
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Gravelle – Rees: A Pareto optimal allocation maximizes a
Paretian swf subject to the production and material balance
constraints
The Paretian swf gives rise to welfare indifference curves with
slopes –W1/W2 < 0
- Pareto efficiency is necessary for Pareto optimality
- Pareto efficiency does not imply Pareto optimality
(NOTE: the above according to definitions in Gravelle - Rees
However the most common definition of Pareto optimality coincides with the definition of Pareto efficiency)
Arrow’s impossibility theorem
• The social ordering in a SWF should be derived from the
individual preference orderings. (A SWF is a function directly on the individuals’ preference orderings, that can
be represented by a swf)
• The SWF should yield transitive social choices
Desirable properties (conditions)
- Unrestricted domain (U)
- Non-dictatorship (D)
- Pareto-principle (P)
- Independence of irrelevant alternatives (I)
Problem:
No SWF exists which satisfies the four conditions, U, P, I and
D, and which can produce a transitive preference ordering
over social states → any process which does yield a social ordering must violate at least one of the conditions
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
The Hicks-Kaldor compensation test
An attempt to compare social situations without the need to
construct a SWF/swf
h
Define for each individual h, CV12 as the compensating variation for a change from situation 1 to situation 2: the amount of
money h would be willing to give up to be in situation 2 rather
than in situation 1 (positive or negative)
∑ CV
h
12
>0 ⇒
the gainers could compensate the losers
h
and situation 2 f situation 1
Critiques
- As the compensation is hypothetical interpersonal value
judgements cannot be avoided
- The Scitovsky paradox (see figure 13.6)
The compensation test may lead to cycles because compensation is not actually paid → only if income effects
are zero can we be sure that there is no paradox
For normal (not inferior) “goods” the problem is that it is
possible that changes in both directions will give a negative result
Example: S1; smoking allowed, S2; smoking forbidden
S1 → S2: Smoker’s CV = -80, non-smoker’s CV = +70
S2 → S1: Smoker’s CV = +60, non-smoker’s CV = -90
Conclusion: The Hicks-Kaldor compensation test favours
status quo!
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
First Theorem of Welfare Economics
If (a) there are markets for all commodities which enter into
production and utility functions and (b) all markets are competitive, then the equilibrium of the economy is Pareto efficient
- All conditions for Pareto efficiency are met in equilibrium
- The Lagrange multipliers from the optimization problem =
= prices (price relations) in equilibrium
ρ1 p1
=
ρ 2 p2
ω1 w1
=
ω 2 w2
Demonstrated for the “two of each” case
(supposed to be price takers)
Consumption choices
u11 p1 u12
MRS = 1 =
= 2 = MRS 212
u 2 p2 u 2
1
21
Supply of inputs
u zh wh
MRS = − h =
ui
pi
h
iz
pi f hi = wh
⇒
(1) (individuals)
f hi =
wh
pi
u zh wh
MRS = − h =
= f hi
ui
pi
h
iz
(2) (firms)
h, i = 1,2
10
(1) + (2)
Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Input use
MRTS
1
21
f11 w1
f12
= 1=
= 2 = MRTS 212
f 2 w2
f2
Output mix
p1 = mc1 =
⇒
w1 w2
= 1
1
f1
f2
p2 = mc 2 =
p1 mc1
f12
f 22
=
= 1 = 1 = MRT21
p2 mc 2
f1
f2
MRT21 =
w1 w2
= 2
2
f1
f2
⇒
Note the error on page
295, expression D.9!
mc1
p
= 1 = MRS 21
mc 2 p2
The fact that the equilibrium of a complete set of competitive
markets is Pareto efficient does not imply that any particular
market economy achieves a Pareto optimal allocation
1) The market economy may not be Pareto efficient
(a) Firms and consumers may not be price-takers
(b) Markets may not be complete
(e.g. no market for clean air, incomplete futures markets and in case of uncertainty incomplete markets
for all states of the world)
(c) Markets may not be in equilibrium
2) Even if the conditions are satisfied this ensures only that the
market allocation is Pareto efficient, not that it is Pareto optimal (remember Gravelle – Rees definition!)
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Second Theorem of Welfare Economics
If all consumers have strictly convex preferences and all firms
have convex production possibility sets, any Pareto efficient
allocation can be achieved as the equilibrium of a complete
set of competitive markets after a suitable redistribution of initial endowments
- Conclusion: It is possible to resolve the conflict between efficiency and equity by intervening to redistribute (by lumpsum transfers) the initial endowments of individuals and then
let the market allocate resources efficiently
Some caveats
(a) Markets are neither complete nor competitive
(b) If preferences and technology are not convex the relative
prices may not support the desired efficient allocation as a
competitive equilibrium. See figures 13.7 and 13.8 (however
note the error in figure 13.8)
(c) Lump-sum transfers are hard to find and non-lump-sum
transfers violate the efficiency requirements as the individuals
are able to alter their tax bills and the subsidies they receive by
changing their behaviour → the set of allocations which can
be achieved by redistribution depends on the original distribution of endowments and does not include all Pareto efficient
allocations achievable by lump-sum transfers. See figure 13.9
The second-best problem and the optimal tax problem
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Market failures
If there is inefficiency it is possible by exchange or production
to make at least one person better off without making anyone
else worse off → Inefficiency implies the existence of mutually advantageous trades or profitable production decisions →
The question of why a particular resource allocation mechanism is inefficient can be rephrased as the question of why
such profitable exchanges or production decisions do not occur
Insufficient control over commodities
- Imperfect excludability
o no individual can acquire exclusive control (common property resources)
o no legal right to exclude or high (infinitive) exclusion costs → potentially beneficial production may
not occur
- Non-transferability
o the owner has not the unrestricted legal right to
transfer use or ownership to any individual on any
terms (maximum or minimum prices fixed by law,
slavery illegal by law)
Information and transaction costs
- Search costs, observation costs, negotiating costs, enforcement costs
Bargaining problems
- Failure of the trading parts to agree upon terms. Often the
result of imperfect competition, (at the equilibrium of a
competitive market there is nothing to bargain about)
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Monopoly
Deadweight loss (figure 14.1)
Potential gains for consumers and the monopolist where the
consumers pay a lump sum to the monopolist to compensate
him for the profit he loses when he decreases price and increases output
Inefficiency persists because consumers and the producer fail
to conclude a mutually satisfactory bargain
• Not able to agree on the division of the gain
• High costs for locating and organizing consumers
• Free-rider problem
• Individual contracts not a solution because of the monopolist’s inability to prevent resale
Externality
Some of the variables which affect one decision-taker’s utility
or profit are under control of another decision-taker
Example: The upstream chemical factory and the downstream
brewery
Consider a reduction in the amount of effluent from the
amount that maximizes the factory’s profit: If the reduction in
the brewery’s costs exceeds the reduction in the chemical factory’s profit there are potential gains from trade and the initial
level of effluent cannot have been efficient → solution to the
problem from Coase (1960) (figure14.2)
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
The Coase Theorem
Bargaining can achieve an efficient allocation of resources
whatever the initial assignment of property rights (figure 14.2)
The initial assignment of the right does however affect the distribution of income (and if income effects ≠ 0 also the final
allocation in case of consumer externalities)
Problems:
• In small number externality situations, there may be failure to agree on the division of the gains
• In large number externality cases we have the same problems as with a monopoly (costs for locating and organizing, free rider problem)
• The legal situation may not be well defined
• The market may not be competitive (one polluter, many
victims)
Pigovian taxes
If parties cannot internalize the externality → regulate by
taxes or subsidies equal to the marginal externality in the efficient situation, a “Pigou tax”
Problem (besides information): Suppose that the parties nevertheless can bargain about the level of the externality → the final allocation will not Pareto efficient (figure 14.3)
A possible solution to the problem is to combine the tax with a
subsidy of the same level. Leads to efficiency provided that
the brewery can do nothing to alter the damage that it suffers
If both parties can influence the magnitude of the effects efficiency requires that both parties mitigate the damage!
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Common property resources
An asset whose services are used in production or consumption and which is not owned by any one individual
Example: A lake in which all members of a community have
the right to fish (figure 14.4). Fish is sold to the competitive
price p and produced by labour time L = ∑Li
q = f ( L ) = f (∑ Li )
qi =
total catch (suppose diminishing returns)
Li
f (L )
L
ith individual’s catch
pq − wL
net social benefit
Social benefit is maximized when
p ⋅ MPL = w
pqi − wLi
Each individual maximizes
As long as profit is positive more individuals will start fishing →
profit will decrease for everyone and this will go on until profit = 0,
that is until in equilibrium
pq = wL
→
p ⋅ APL = w
• Free access leads to overfishing if there are diminishing
returns
• Free access leads to underinvestment
o No account for other users’ gains
o Investment by other users is a substitute for own investment
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Possible solutions:
• Voluntary agreement
Any individual will always find it profitable to break the
agreement
• Divide the lake among the fishermen → exclusive fishing
rights to part of the lake
Policing and enforcing costs
• Ownership to one individual
Efficient, provided that the final fish market remains competitive
Public goods
• Non-rival goods
o Optional, and non-optional
Necessary conditions for efficiency
Example: (figure 14.5)
Max
u1 ( x1 , q )
s.t . u 2 ( x2 , q ) = u 2
Results: At q*
x1 + x2 ≤ x
f ( x , q ) ≤ PP
MRS 1xq + MRS xq2 = MRT xq
Conditions unlikely to be satisfied in a market economy:
• Many public goods are non-excludable → free rider
problem, production too small
• As the opportunity cost of consumption is zero no one
should be excluded from consuming a non-rivalry good
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Lindahl equilibrium
Suppose a non-excludable public good
An efficient solution will be reached if each individual pays a
price equal to her own personal MRS → they will all agree on
the amount of the public good and the sum of prices will equal
marginal cost
The above is the background to Lindahl’s solution to the problem (see figure 14.6 for two consumers)
• The planner announces a set of tax prices that add up to
marginal cost
• Consumers say how much they would like to consume at
the announced tax prices
• If demands are not equal the planner adjusts tax prices
• The Lindahl equilibrium is reached when consumers report the same demand → efficient solution provided that
consumers are honest
However, the process gives the consumers an incentive to misrepresent their demands
• Suppose individual 2 is honest and taking that into account individual 1 maximizes her utility → individual 1
will understate her demand in order to decrease her tax
price (acting like a monopsonist) → production will be
too small and the process will lead to a welfare loss, (figure 14.7)
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
The theory of the second best
How should the government act in an economy in which there
is market failure?
• Nationalization or regulation of monopolies, correction of
externalities, provision of public goods
In theory possible to reach a first-best optimum but in the real
world there exist non-trivial restrictions on the set of policy
instruments or the set of agents whose behaviour can be influenced
• Example 1: The government has nationalized a monopoly, but there is another private profit-maximizing monopoly → if the government can not affect the private
monopoly directly the second-best policy is to choose a
price above marginal cost for the nationalised monopoly
• Example 2: A road exists on which no toll is charged.
The road is however severely congested leading to a
“price” (AC) < MC. The public agency constructs a road
bridge as a substitute for the existing road. What is the
optimal capacity and toll for the public road? (Figures
14.8 and 14.9)
Given the restriction that no toll can be charged on the existing
road the toll for the new bridge should be set below LMC
Optimal condition for the toll, p0:
( p* − p 0 )
∂x
∂y
= − ( MC − v ( y ))
∂p
∂p
Marginal welfare
loss on bridge
Marginal welfare
gain on existing road
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Föreläsningsunderlag för Gravelle-Rees. Del 3. Thomas Sonesson
Government action and government failure
• Traditional welfare economic approach: In case of market
failure the government acts in order to achieve Pareto efficiency (by regulations, laws, subsidies, taxes, own production and provision of services etc.)
• Objections: It must be established that the government is
both willing and able to act in this way → we need a
positive theory of government decision making, taking
into account:
o Non-altruism
o Information costs
o Actual decision rules (e.g. the majority rule)
Conclusion: Market failure does not imply the necessity of
government action, and likewise government failure does not
imply that the scope of market allocation should be extended
→ you have to compare the way different institutions allocate
resources in each case
Example: The one size bridge. Monopolistic supply versus
public provision. Compare decisions on construction and use
(price) of the bridge (figure 14.10)
(a) Political authorities want to secure votes to stay in charge
(b) Demand estimation: Managers in a private firm are more
risk-avert. Public sector decision makers are less punished for
an inefficient decision to build the bridge
(c) and (d) A public bridge may have higher than minimum
construction costs/marginal running costs because of lack of
incentives for public officials to ensure that costs are minimized
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