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FUR XII 2006, Rome 2
IntroductionMotivation
Optimal public policy in the presence of Knightian uncertainty and ambiguity aversion
Approach normative: how should a regulator behave?
Application To find the best tariff regulation of a monopolist in the presence of asymmetric information on his production costs
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OutlineOptimal regulation models:
Loeb and Magat (1979): Non Bayesian approachBaron and Myerson (1982): Bayesian approach
Decision criteria under ignorance:Insufficient reason (Laplace), Maximin (Wald), Leximin (lexicographic maximin, Maskin)
Decision criteria under ambiguity:simple capacity linear combination of Max EU and Maximin (Gilboa, 1988)complex capacity intermediate case between Max EU and leximin
Optimal regulation policy with different decision criteria
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The L-M modelC(q) = K + cq, (p,T,c) = (p – c)q(p) – K + T Objective: to implement the first best tariff by means of a decentralization of the tariff’s choiceOptimal policy: T = CS (p)Result: pLM = cMain limits:
The firm gains a rent equal to all the surplus created in the market
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The B-M modelSetting
C(q) = K + cq, where c c*,c* , F(c)(p,T,c) = (p – c)q(p) – K + T W(p,T,c) = CS(p) – T + (p,T,c)
Resolution by means of the Revelation Principle
Optimal tariff function: pBM(c)= c + (1-)F(c)/f(c), c c*,c* F(c)/f(c) must be non decreasing (suff.
condition)
)c(p
max 1
c
cSS( p( c )) f ( c ) ( )q( p( c ))F( c ) dc
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Comments on B-M modelTrade off between allocative and distributive efficiencypBM = pLM iff = 1Limit of the Bayesian approach:
implicit hypothesis regarding ambiguity: absence or neutral attitude
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Decision criteria under ignorance
Principle of insufficient reason:the decision criterion weighs equally every potential consequence
Maximin principle:the decision criterion put all the weight on the worst consequence
Leximin Principle:it functions as an iterated maximinwithin every set of consequences, the decision weight is concetrated on the worst one
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Ambiguity and capacitiesX = x1, … xN
b:2X[0,1] where:1. b() = 0 and b(X) = 12. E, F 2X: E F b(E) b(F), 3. E, F 2X: b(E F) b(E)+ b(F) –
b(EF);
Choquet expected utility:U(b) = u(xi) [b(Ei) - b(Ei-1)]
Note: p(c) c dW/dc 0
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Simple capacitybs(E) =(1 (E) E 2X, E X bs(X) = 1
where: () is an additive probability [0,1]
Gilboa (1988):CEU(bs) = (1 )EU(x; ) + min u(x)
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Complex capacity
bc(E) =
where: () is an additive probability [1,)
Property 1: = 1, n = 1,… N.
N |E|
N
( E )
i
in1
nlim
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Optimal regulation with ambiguous beliefs/1
Result 1: pEU-M(c)= pBM(c),c c*,c° pEU-M(c)= pBM (c°), c c°,c*.
if =0, pEU-M(c)= pBM(c),c c*,c*; if 1, pEU-M(c°) c*.
The consequence of ambiguity aversion consists only in limiting the maximum possible tariff The aim of this policy is to put a threshold to the minimum achievable social welfare: prudent attitude
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Optimal regulation with ambiguous beliefs/2
pEU-L(c) = c + (1 )
if pEU-L(c) = pBM(c) c c*,c*; if , pEU-L(c) pLM(c) c c*,c*.
The consequence of ambiguity aversion consists in a tariff closer to the first best oneThis policy is characterized by a pessimist attitude
1F( c )
f ( c ) F( c )lnc c
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A graphical interpretation
0
1
2
3
4
5
6
7
8
1 2 3 4 5
cost level
op
tim
al p
rice pBM
pLM
pEU-M
pEU-L
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ConclusionsGeneralization of the BM’s model to an uncertain environment applying two different decision criteria Combination of EU and Maximin:
the optimal policy is modified by the presence of a ceiling to the maximum tariffprudent attitude
Intermediate criterion between EU and Leximin:the optimal policy approaches to the first best tariffpessimist attitude
Further extension: minimax regret criterion (Lopez Cuñat, 2000)Hurcwitz criterion and neo-capacities (Eichberger and Chateneuf 2005)
Can we define different types of ambiguity aversion?
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Thank you for your attention!
