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Lecture notes

MT2 Welfare Economics and Social Choice Notes

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These notes were prepared based on the lectures and supplemented by information from textbooks and tutorials where parts of the lecture were unclear. Graphs, equations, and bullet-point explanations included. Prepared by a first class Economics and Management student for the FHS Microeconomics pape...

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  • June 27, 2024
  • 15
  • 2022/2023
  • Lecture notes
  • Simon cowan
  • Mt2 welfare economics and social choice
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MT2 Microecons (Welfare Economics and Social Choice)
What we'll cover
 Competitive Equilibrium as a normative theory
 Definition of welfare and welfare optimization

 The Two Fundamental Theorems of Welfare Economics
 Social choice and voting
o How do we aggregate ordinal preferences?
o How do we implement a social outcome?
o Can we induce citizens to tell the truth?


Lecture 4: Competitive Equilibrium and Welfare
Introduction: UPF, Pareto efficiency, and Kaldor-Hicks improvement
 Pareto efficiency
o Consider two allocations 𝒙 and 𝒙′. Allocation 𝒙 Pareto-dominates allocation 𝒙′ if
everyone weakly prefers 𝒙 to 𝒙′ and at least one agent strictly prefers 𝒙 to 𝒙′.
 Formally: 𝒙𝑃𝒙′ if 𝒙 ≽𝑖 𝒙′ for all agents 𝑖 and 𝒙 ≻𝑖 𝒙′ for some 𝑖
o An allocation is Pareto-efficient if it is not Pareto-dominated:
 No one can be made better off without making someone worse off
 Weak criterion
 Not all outcomes can be Pareto-compared
 e.g., (1, 1, 1) vs. (2, 2, 1−𝜀). Better in two, worse in one: cannot
compare. Can only be compared when one dominates another.
 Says nothing about inequality, only about absence of free lunches
 e.g., (3,0,0) vs. (1, 1, 1) are both Pareto efficient, but one is more equal
than the other
 Utility possibility frontier (UPF)
o The utility possibility frontier describes the utility of agents when we reallocate a fixed
level of resources 𝒙.
 𝑼 = {𝒖(𝒚) such that Σ𝑖𝒚𝑖 = Σ𝑖𝒙𝑖}
 Utility profile of individuals at 𝒙: 𝒖(𝒙)=(𝑢1(𝒙), 𝑢2(𝒙), 𝑢3(𝒙), …)




o
o The graph shows the possibility sets of {𝑢1, 𝑢2} given different allocations of fixed total
amount of good 𝑋
 Suppose 𝑢𝑖 =𝑥𝑖𝛽 and there’s a single good in fixed total amount 𝑋, so 𝑥1 + 𝑥2
=𝑋
 𝑢1 = 𝑥1𝛽 so rearrange to 𝑥1 = 𝑢1(1/𝛽).

,  Hence, 𝑢2 = (𝑋 − 𝑥1)𝛽 = (𝑋 − 𝑢1(1/𝛽))𝛽
 UPF is Concave if 𝛽<1. Linear if 𝛽=1.
 UPF, Pareto efficiency, and Kaldor-Hicks improvement




o
o Points inside the UPF are not Pareto-efficient, while points on the UPF (the green line)
are Pareto efficient
o Top right rectangle of each point is Pareto dominates it
o 𝒙′ Pareto-dominates 𝒙 because
 𝑢1(𝒙′ ) > 𝑢1 (𝒙) and 𝑢2(𝒙′) > 𝑢2(𝒙)
o 𝒙′′ cannot be Pareto-compared to 𝒙′ because
 𝑢1(𝒙′ ) > 𝑢1 (𝒙'') but 𝑢2(𝒙′') > 𝑢2(𝒙')
 Some points cannot be Pareto-compared (neither Pareto-dominates the other)
o Kaldor-Hicks improvement
 𝒙′′ does not Pareto dominate 𝒙
 𝒙′′ does not Pareto dominate 𝒙′
 But since by definition Σ𝑖 𝒙𝑖′′ = Σ𝑖 𝒙𝒊′, we could feasibly reallocate resources
from 𝒙′′ to achieve 𝒙′ that Pareto-dominates 𝒙!
 A re-allocation is a Kaldor–Hicks improvement if those that are made better off
could hypothletically compensate those that are made worse off and lead to a
Pareto-improving outcome.
 The compensation does not actually have to occur and thus, a Kaldor–Hicks
improvement can leave some people worse off.
 A situation is said to be Kaldor–Hicks efficient, or equivalently is said to satisfy
the Kaldor–Hicks criterion, if no potential Kaldor–Hicks improvement from that
situation exists.

Social welfare function
 A social welfare function 𝑊(𝒖) helps to tell us what the best point on the UPF is:
o Concern for the worst off (red): 𝑊(𝒖) = min { 𝑢𝑖(𝒙)}
o Utilitarian (blue): 𝑊(𝒖) = ∑𝑖 𝑢𝑖(𝒙)

 Gradient = -1. 1 unit of u1 must be compensated by 1 unit of u2
o Inequality-averse (green):
 Alpha weighs the utilities. To be inequality averse, you could weigh the utility of
lower utility individuals more

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