Social Choice: Lecture 15

Social Choice: Lecture 15

Ram Singh

Course 001

October 29, 2014

Social Choice I

Question

How to choose from the feasible set of alternatives?

Do societies have preference relations similar to the ones assumed for individuals?

Is Pareto criterion helpful here?

What are the other approaches possible in a social context?

Scitovsky Criterion I

Definition

xis Scitovsky superior toy, i.e.,xSyif

xKy but

∼yKx

xKyimplies there existsz∈S(x)such thatzPy. That is, (∀i ∈N)[zRiy]

(∃j∈N)[zP_jy]

But, there is not∈S(y)such that

(∀i ∈N)[tRix]

(∃j ∈N)[tP_jx]

Scitovsky Criterion II

Definition

All social states/alternatives are accessible from each other if (∀x,z,z)[z∈S(x)⇒z∈S(y)]

Proposition

If all social states/alternatives are accessible from each other thenxSyif and only ifxis P.O butyis not P.O

Proposition

If all social states/alternatives are accessible from each other thenxKyif and only ifyis not P.O.

Samuelson Criterion

Definition

xis Samuelson superior toy, i.e.,xS¯yif for anyz∈S(y) xKz

That is, for anyz∈S(y), there existsw∈S(x)such thatwPz, i.e.,

(∀i ∈N)[wR_iz]

(∃j ∈N)[wPjz]

Social Choice Criteria: Compared

Consider

x= (50,100,150) ,i.e.,

3

X

i=1

xⁱ =300

y= (90,90,90) ,i.e.,

3

X

i=1

yⁱ =270

z= (80,250,250) ,i.e.,

3

X

i=1

zⁱ =580

Question

Which of the above alternatives is efficient according to Pareto, Rawlsian, Kaldor-Hicks Efficient, Scitovsky and Samuelson criteria?

Defining SWF I

Let

W(u¹,u², α₁, α₂) =α₁u¹(x₁,y₁) +α₂u²(x₂,y₂)

x1,x2max,y1,y2,x,yW(u¹,u², α₁, α₂) (1)

s.t.

x₁+x₂ = f(l_x,t_x) y1+y2 = g(ly,ty)

l_x+l_y = ¯l tx+ty = ¯t

Let(x₁^∗,x₂^∗,y₁^∗,y₂^∗,x^∗,y^∗), wherex^∗=f(l_x^∗,t_x^∗)andy^∗=f(l_y^∗,t_y^∗)be a solution to (1).

Proposition

(x₁^∗,x₂^∗,y₁^∗,y₂^∗,x^∗,y^∗)solves (1) only if(x₁^∗,x₂^∗,y₁^∗,y₂^∗)is Pareto optimum.

Wealth Maximization I

Definition

When allocations have money equivalent, i.e., when allocations can be described in money terms, Kaldor-Hicks Efficient allocation is wealth maximizing.

Proposition

When allocations have money equivalent, a wealth maximizing allocation is Pareto efficient.

Consider

two individuals:i =1,2

one FOPy to be used to produce wealth/income

Wealth Maximization II

Consider the OP for wealth maximization:

maxy {w¹(y) +w²(y)} (2)

FOC : dw¹(y)

dy +dw²(y)

dy =0 (3)

Next, consider the OP for Pareto or Kaldor efficiency:

maxy,t {u¹(w¹(y)−t)} (4) subject to : u²(w²(y) +t)≥¯u² (5) Using Lagrangian technique, we get FOCs:

Wealth Maximization III

u⁰¹dw¹(y)

dy +λu⁰²dw²(y)

dy = 0 (6)

u⁰¹+λu⁰² = 0 (7)

the above FOCs give us

dw¹(y)

dy +dw²(y)

dy =0 (8)

Wealth Maximization IV

Problems:

Every aspect of individual welfare cannot be monetized

When when money equivalent is possible, it will depend on relative prices of goods

Relative prices depend on demand

Demand depends on value of endowments, which in turn depends on relative prices

So, how can we determine initial endowments?