Welfare Economics And Social Choice

Question 1

When is an allocation Pareto efficient?

Answer 1

An allocation is pareto-efficient if it is not Pareto-dominated.

Allocation x Pareto-dominates allocation x’ if everyone weakly prefers x to x’ and at least one agent strictly prefers x to x’

Question 2

What is the utility possibility frontier?

Answer 2

Fix a level of resources at 𝒙.
Consider utility profile of individuals at 𝒙
𝒖(𝒙) = (𝑢1(𝒙), 𝑢2(𝒙), 𝑢3(𝒙), … )

The utility possibility frontier describes the utility of
agents when we reallocate these resources
𝑼 = {𝒖(𝒚) such that Σ𝑖𝒚𝑖 = Σ𝑖𝒙𝑖}

Question 3

When is a UPF strictly concave or linear?

Answer 3

Suppose 𝑢𝑖 = 𝑥𝑖
𝛽 and there’s a single
good with a fixed total amount 𝑋, so
𝑥1 + 𝑥2 = 𝑋

𝑢2(𝑢1) = (𝑋 − 𝑥1(𝑢1))𝛽
= (𝑋 − 𝑢1
^1/𝛽) ^𝛽

Strictly concave if 𝛽 < 1.
Linear if 𝛽 = 1.

Question 4

When can two allocations not be Pareto-compared?

Answer 4

If they are both on the UPF/both Pareto-efficient

Question 5

What is a Kaldor-Hicks improvement?

Answer 5

not a pareto improvement as one is made worse off but one could compensate two and be better off than before

Question 6

What is a social welfare function and examples?

Answer 6

Social Welfare function W(u) determines best point on utility possibility frontier.

Concern for the worst off: 𝑊 (𝒖) = min{𝑢1, 𝑢2}
Utilitarian: 𝑊(𝒖) = 𝑢1 + 𝑢2
Inequality-averse: 𝑊 𝒖 = 𝑢1^𝛼 * 𝑢2 ^1−𝛼

Question 7

What is society’s optimization problem?

Answer 7

Fix a resource constraint Σ𝑖𝒙𝑖
Derive the utility possibility frontier 𝑼
Maximize welfare 𝑊(𝒖) subject to 𝑼

Question 7

How do you find a Pareto efficient allocation?

Answer 7

Fix Bob’s utility level at u (with bar)
and maximise Alice’s utility
subject to the resource
constraints and Bob
maintaining his utility

max 𝑢𝑎(𝒙) subject to
Σ𝑖𝒙𝑖 = Σ𝑖𝒘𝑖
and 𝑢𝑏(𝒙) ≥ u (with bar)

Question 8

What is the set of Pareto efficient allocations

Answer 8

𝑀𝑅𝑆𝑎 = 𝑀𝑅𝑆𝑏

Use endowments/resource constraints to give x2a as a function of x1a along the contract curve

Question 9

What is the first fundamental theorem of welfare economics?

Answer 9

If (𝒙∗, 𝒑∗) is a competitive equilibrium in the exchange economy, then 𝒙∗ is Pareto-efficient

Question 10

What is the proof of the first fundamental theorem of welfare economics?

Answer 10

• Uses the fact that preferences are monotone.
• Note that if an allocation is strictly preferred to another by one agent, say
  a, and weakly preferred by all others, then another feasible allocation can
  be found that is strictly preferred to the original allocation by all agents.
  Divide some of the gains for a between all the other agents.
• Proof is by contradiction. Two agents. Suppose 𝒙∗ is not Pareto efficient
  so there is an allocation 𝒚 such that
  y1a + y1b = w1a + w1b
  y2a + y2b = w2a + w2b
  𝑢𝑖(𝑦𝑖1, 𝑦𝑖2) > 𝑢𝑖(𝑥𝑖1, 𝑥𝑖2) for each agent

As 𝑦𝑖 is preferred it must cost more at the equilibrium prices:
𝒑∗ ∙ 𝒚𝒊 > 𝒑∗ ∙ 𝒙𝒊*
As (𝒙∗, 𝒑∗) is a competitive equilibrium: 𝒑∗ ∙ 𝒙𝒊∗ = 𝒑∗ ∙ 𝒘𝒊

Add the resulting inequalities for the two agents. Creates a contradiction with feasibility equations. So x* is pareto efficient

Question 11

What is the second fundamental theorem of welfare economics?

Answer 11

Suppose 𝒙∗ > 𝟎 is a Pareto-efficient
allocation in the exchange economy and that preferences are convex, continuous, and monotonic.
Then (𝒙∗, 𝒑∗) is a competitive equilibrium for the initial endowments with 𝒑∗ ∙ 𝒘 = 𝒑∗ ∙ 𝒙∗

Question 12

What are consequences of the second fundamental theorem of welfare economics?

Answer 12

• Any Pareto efficient allocation can be implemented
  as a market outcome.
• All the caveats for 1st FWT apply but even more
  stringent.
• Large redistribution of resources would be required.
• Lump sum redistribution required.
• Huge informational demands on the planner.

Question 13

What is a preference aggregation rule?

Answer 13

map 𝐹 assigning to each preference profile (≿1, … , ≿𝑛) a preference ordering ≿∗ for society:
𝐹 ≿1, … , ≿𝑛 = ≿∗

• In a cardinal world, SWF given by 𝑊(𝑢1, … , 𝑢𝑛) (e.g. Σ𝑖=1
  𝑛 𝛼𝑖𝑢𝑖)
• In general, we want ≿∗ to be complete and transitive.
• Often we use a voting procedure to aggregate opinions.

Question 14

What is Arrow’s impossibility theorem?

Answer 14

Suppose there are at least three social states. Then, there is no
preference aggregation rule 𝐹 which satisfies
* Unrestricted domain
* Pareto principle
* Non-dictatorship
* Independence of irrelevant alternatives.
and generates a complete and transitive preference ordering for
society.

Question 15

What is Black’s (1948) theorem regarding single-peaked preferences?

Answer 15

If alternatives are one-dimensional and voters have single-
peaked preferences, then majority rule induces a transitive
social preference ordering and selects the median
alternative over others.

Question 16

What is the median voter theorem?

Answer 16

If citizens’ preferences are single-peaked, and if there are two
candidates who can commit in advance to policies and only
care about winning, then there is a unique equilibrium
𝑝1∗ = 𝑝2∗ = 𝑏𝑚

In other words, both candidates pick the policy preferred by the median voter, whose bliss point is 𝑏𝑚.

Question 17

What is strategy-proofness? And related impossibility theorem (Gibbard, 1973)

Answer 17

Suppose there are at least three social states.
Then there is no social choice function which is
* strategy-proof, and
* non-dictatorial.

Implication:
* Every election rule can be manipulated (by voters…)
* Agents can have incentives not to vote according to their true
rankings.
* Requires full knowledge about preferences of others
* Some voting systems “harder” to manipulate than others