Individual and Social Choices
Ram Singh
Microeconomic Theory
Lecture 16
Preferences and Choices I
Let
Xbe the set of alternatives
Ri be the ‘weak’ preference relation for individuali, defined overX; i =1, ...,n
Pi be the strict preference relation for individuali Rbe the set of individual preference relations
Obe the set of individual preference relations that are orderings;O⊂R. (R1, ...,Rn)∈Rnbe a profile of preference relations - one for each individuals. That is,
Rn={(R1, ...,Rn)|Ri ∈Rfor eachi=1, ...,n}
Rbe a ‘weak’ Social preference relation;R∈R
Preferences and Choices II
Assumption
Every social preference relationRi
has strict preference relationPi and indifference preference relationIi associated with it.
Pi andIi are such that: For allx,y ∈X
xPiy ⇔ xRiy andv(yRix) xIiy ⇔ xRiy andyRix
Choice Set: Take anyS⊆X. The choice set generated by the preference relationRdefined over the setSis given byC(S,R), where
x ∈C(S,R)if and only if(∀y ∈S) [xRy],i.e.,
Preferences and Choices III
C(S,R) ={x|(∀y ∈S) [xRy]}
Definition
LetS⊆X. An alternativex is a ‘best’ elements ofSiff (∀y ∈S)[xRy]
A setC(S,R)is the set of ‘best’ elements ofSiff [x ∈C(S,R)]⇔(∀y ∈S)[xRy]
Preferences and Choices IV
Definition
LetS⊆X. An alternativex is a Maximal elements ofSiff:
v(∃y ∈S)(yPx)]
A setM(S,R)is the set of Maximal elements ofSiff: For allx ∈S, [x ∈M(S,R)]⇔[v(∃y ∈S)(yPx)]
Suppose
∼xRyand∼yRx.
So,M(S,R) ={x,y}. ButC(S,R) =∅.
Therefore,x ∈M(S,R)does not mean that for ally ∈S,xRyholds.
Preferences and Choices V
Proposition
For any giveS⊆Xand preference relation R, C(S,R)⊆M(S,R).
Definition
A preference relation is a quasi-ordering if it is reflexive and transitive.
Proposition
IfS⊆Xis finite and and preference relation R is quasi-ordering, then M(S,R) is non-empty.
Preferences and Choices VI
LetS={x1, ...,xn}. Leta1=x1,
a2=
x2, ifx2Px1 a1, otherwise.
aj+1=
xj+1, ifxj+1Paj
aj, otherwise.
You can verify thatanis a maximal element.
Social Choice Rules (SCR) I
Assumption
Every social preference relationR
has strict preference relationPand indifference preference relationI associated with it.
P andIare such that: For allx,y ∈X
xPy ⇔ xRy andv(yRx) xIy ⇔ xRy andyRx
Assumption
We assume individual preferences are ‘orderings’, i.e., are reflexive, complete and transitive. That is, for alli =1, ..,n,Ri ∈O.
Social Choice Rules (SCR) II
Definition
A SCR is a function
f :Rn7→R, such that,
(∀(R1, ...,Rn)∈Rn)[f(R1, ...,Rn) =R∈R].
Definition
A SCRf is decisive iff∀(R1, ...,Rn)∈Rn, the social preference relation generated byf is complete, i.e., iff∀(R1, ...,Rn)∈Rn,f(R1, ...,Rn) =Ris complete.
Definition
A SCR is rational if∀(R1, ...,Rn)∈Rn, the social preference relation generated byf, i.e.,f(R1, ...,Rn) =R, is an ordering.
Pareto Criterion as SCR I
Definition
Consider the following preference relations: Forx,y ∈X, xRy¯ ⇔ [(∀i ∈N)[xRiy]]
xP¯y ⇔ [xRy¯ & v(yRx¯ )]
xI¯y ⇔ [xRy¯ &yRx¯ ]
Proposition
RelationR¯ is a quasi-ordering. That is, it is reflexive and transitive.
Pareto Criterion as SCR II
Definition
The Pareto Criterion is the SCR iff xRy ⇔ xRy¯ ,
xPy ⇔ [xRy¯ & v(yRx¯ )]
xIy ⇔ [xRy¯ &yRx¯ ]
Proposition
If Pareto Criterion is used as a SCR, then for any finiteS⊆Xthe set of maximal elements is non-empty.
Pareto Criterion as SCR III
Definition
The SCR is Pareto inclusive, i.e., satisfies the Pareto Criterion if: For all x,y ∈X
(∀i ∈N)[xRiy],i.e.,xRy¯ ⇒ xRy xRy¯ and vyRx¯ ⇒ xPy Note:
If Pareto Criterion is used as the SCR, then the SCR is Pareto inclusive However, a Pareto inclusive SCR can (will) be different from the Pareto Criterion
Pareto Criterion as SCR IV
Proposition
Pareto Criterion is a decisive SCR iff
(∀x,y ∈X)[(∃i ∈N)[xPiy]⇒(∀j ∈N)[xRjy]]
Suppose,∃i∈Nsuch thatxPiy, and at the same time∃j ∈Nsuch thatyPjx. In that case, we have
vxRy¯ and v(yRx¯ ),i.e., Therefore, the condition is necessary.
SCRs: Desirable Features I
Take anyD⊆On. Definition
Condition O: A SCR satisfies condition O, iff :D7→O. That is, (∀(R1, ...,Rn)∈D)[f(R1, ...,Rn) =R∈O].
We call such a SCR a Social Welfare Function (SWF).
Definition
Condition U: A SCRf satisfies condition of ’unrestricted domain’, if its domain isOn. That is,
f generates a social preference relation for every possible profile of individual preferences.
SCRs: Desirable Features II
Definition
Condition P: A SCRf satisfies condition of ’Weak Pareto Principle’, if (∀x,y ∈X)(∀i ∈N)[xPiy ⇒xPy].
LetS={x,y}. Consider
ANY two profiles of individual orderings, say(R1, ...,Rn)and(R10, ...,Rn0).
Letf(R1, ...,Rn) =Randf(R10, ...,Rn0) =R0. Now, condition I implies the following:
(∀i ∈N)[xRiy ↔xR0iy] ⇒ (xRy iffxR0y)and(yRx iffyR0x),i.e., (∀i ∈N)[xRiy ↔xR0iy] ⇒ C(S,R) =C(S,R0).
SCRs: Desirable Features III
Definition
Condition I: Take anyS⊆X, and ANY two profiles of individual orderings, say (R1, ...,Rn)and(R10, ...,R0n). Letf(R1, ...,Rn) =Randf(R10, ...,Rn0) =R0. A SCRf satisfies condition of ‘independence of irrelevant alternatives’ if the following holds:
(∀x,y ∈S)(∀i ∈N)[xRiy ⇔xRi0y]⇒C(S,R) =C(S,R0)
Definition
Condition D: A SCRf satisfies condition of ’non-dictatorship’, if there is NO individuali ∈Nsuch that
(∀x,y ∈X)[xPiy ⇒xPy].
