Lecture 7: Pareto Optimality | Summary and Q&A

TL;DR

Pareto optimality is achieved by maximizing a weighted sum of expected utilities, while risk sharing is determined by the aggregate social endowment and individual risk aversion.

Key Insights

• 💄 Pareto optimality ensures that no one can be made better off without making someone else worse off.
• 🚙 The utility possibilities set represents all feasible utility profiles for households in an economy.
• 🚙 The Pareto frontier represents the boundary of the utility possibilities set and consists of utility vectors that cannot be improved upon without making someone worse off.
• 🍹 The planner's problem involves maximizing a weighted sum of utilities subject to resource constraints.
• 😥 The supporting hyperplane theorem guarantees the existence of a hyperplane that is tangent to a convex set at a given point, allowing for the determination of the weights that make a Pareto optimal allocation a solution to the planner's problem.
• ✳️ The solutions to the planner's problem can be used to determine optimal allocations of risk.
• 🚙 Different utility functions and risk aversion levels can result in different optimal allocations.

Transcript

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Questions & Answers

Q: What is Pareto optimality?

Pareto optimality is a state where an allocation is considered optimal if it cannot be improved upon without making someone worse off.

Q: What is the utility possibilities set?

The utility possibilities set represents all possible utility vectors for households in an economy that can be attained by a feasible allocation.

Q: What is the Pareto frontier?

The Pareto frontier is the boundary of the utility possibilities set, consisting of utility vectors that cannot be improved upon without making someone worse off.

Q: What is the planner's problem?

The planner's problem involves maximizing a weighted sum of utilities subject to resource constraints, and the solution represents a Pareto optimal allocation.

Q: What is the supporting hyperplane theorem?

The supporting hyperplane theorem guarantees the existence of a hyperplane that is tangent to a convex set at a given point, which can be used to find the weights that make a Pareto optimal allocation a solution to the planner's problem.

Summary & Key Takeaways

• Pareto optimality is a concept where an allocation is considered optimal if there is no other allocation that makes at least one person better off without making someone else worse off.

• The utility possibilities set represents all utility profiles for households in an economy that can be attained by a feasible allocation.

• The Pareto frontier is the boundary of the utility possibilities set, consisting of utility vectors that cannot be improved upon without making someone worse off.

• The planner's problem involves maximizing a weighted sum of utilities subject to resource constraints, and the solution to the problem represents a Pareto optimal allocation.

• The supporting hyperplane theorem guarantees the existence of a hyperplane that is tangent to a convex set at a given point, which can be used to find the weights that make a Pareto optimal allocation a solution to the planner's problem.