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) 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 If excess demand is positive add to the initial price some multiple of the excess demand (increase price) If excess demand is zero let the new price equal the old price 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 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 8
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! 9
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 12
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) 13
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! 15
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 16
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 17
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 19
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 20