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Understanding Optimal Choice: Exploring Preferences and Demand Functions

This document delves into the intricacies of consumer choice theory, focusing on how utility functions and preferences dictate optimal choices. We analyze abstract parameters through the lens of Cobb-Douglas utility functions, outlining their significance in determining demand functions. The study encompasses four types of preferences: Cobb-Douglas, Perfect Complements, and Perfect Substitutes, examining the conditions for interior versus corner solutions. Additionally, we provide geometric interpretations and practical calculations relevant to demand under varying conditions.

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Understanding Optimal Choice: Exploring Preferences and Demand Functions

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  1. L05 Choice

  2. Problem: • We know preferences (utility function) and • We want to know optimal choice

  3. Choice $ $ $ $ $ $ $ $ $ $

  4. Choice: geometric solution x2 x1

  5. Abstract approach • In the example we were given we found demands - two numbers • Now we use abstract parameters • we find demand functionsNow we • 4 types of preferences

  6. Abstract Cobb Douglass Function • Cobb Douglass utility functions and are equivalent in terms of preferences

  7. Abstract Cobb Douglass Function

  8. Magic (Cobb-Douglass) formula Parameters:

  9. Cobb-Douglas: Summary Utility function: or Solution: Shares of income

  10. A) Let and B) Let and

  11. Interiority Cobb – Douglass (always interior solution)

  12. Cobb- Douglass preferences x2 x1

  13. SOH (Perfect Complements)

  14. SOH (Perfect Complements)

  15. Perfect Complements (SOH) Interior or corner solution?

  16. Is solution always interior? • Not necessarily • Even with well behaved preferences we might have a corner solution • Example: Perfect Substitutes

  17. Perfect substitutes $ $ $ $ $ $ $ $ $ $

  18. Perfect Substitutes x2 x1

  19. Magic (Substitutes) Formula

  20. Choice $ $ $ $ $ $ $ $ $ $

  21. Is solution interior? • Hence demand and • Geometric interpretation • How to solve for corner solution? • Find a buddle using standard conditions • If some then in optimum

  22. Choice: Calculation

  23. In Practice • Cobb-Douglass, Perfect Complements? • Quasilinear ? • Perfect Substitutes?

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