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...
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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