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Choice

Choice. Budget constraint Indifference curves More preferred bundles. Q: What is the Optimal Choice?. Optimal choice: indifference curve tangent to budget line. Does this tangency condition necessarily have to hold at an optimal choice?.

denise-dennis
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Choice

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  1. Choice

  2. Budget constraint Indifference curves More preferred bundles Q: What is the Optimal Choice?

  3. Optimal choice: indifference curve tangent to budget line. Does this tangency condition necessarily have to hold at an optimal choice? A: Optimal Choice is X

  4. Perfect Substitutes

  5. Perfect Complements

  6. Q: Is Tangency Sufficient?

  7. What is the General Rule? If: • Preferences are well-behaved. • Indifference curves are “smooth” (no kinks). • Optima are interior. Then: Tangency between budget constraint and indifference curve is necessary and sufficient for an optimum.

  8. A way to avoid multiplicity of optima, is to assume strictly convex preferences. This assumption rules out “flat spots” in indifference curves. Multiple Optima

  9. Economic Interpretation • At optimum: “Tangency between budget line and indifference curve.” • Slope of budget line: • Slope of indifference curve: • Tangency:

  10. At Z: At Y: Interpretation

  11. Tangency with Many Consumers • Consider many consumers with different preferences and incomes, facing the same prices for goods 1 and 2. • Q: Why is it the case that at their optimal choice the MRS between 1 and 2 for different consumers is equalized?

  12. Tangency with Many Consumers • A: Because if a consumer makes an optimal choice, then: • Implication: everyone who is consuming the two goods must agree on how much one is worth in terms of the other.

  13. Indifference curves of consumer 1: Indifference curves of consumer 2: Budget line: Tangency with 2 Consumers

  14. Finding the Optimum in Practice: a Cobb-Douglas Example • Preferences represented by: • Budget line:

  15. Finding the Optimum in Practice: a Cobb-Douglas Example Mathematically, we would like to: such that

  16. Finding the Optimum in Practice: a Cobb-Douglas Example • Replace budget constraint into objective function: • New problem:

  17. Finding the Optimum in Practice: a Cobb-Douglas Example • New problem: • First-Order Condition:

  18. Finding the Optimum in Practice: a Cobb-Douglas Example • First-Order Condition: • Rearranging:

  19. Finding the Optimum in Practice: a Cobb-Douglas Example • First-Order Condition: • Solve for : • Expenditures share in 1:

  20. Finding the Optimum in Practice: a Cobb-Douglas Example • Q: How do I find ? • A: Use the budget constraint:

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