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Utility

Utility. The MRS is the slope of the indifference curve at a point MRS=derivative of indifference curve. Marginal Rate of Substitution. Interpretation of MRS. The MRS measures the rate at which the consumer is willing (i.e., indifferent) to substitute one good for the other.

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Utility

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  1. Utility

  2. The MRS is the slope of the indifference curve at a point MRS=derivative of indifference curve Marginal Rate of Substitution

  3. Interpretation of MRS • The MRS measures the rate at which the consumer is willing (i.e., indifferent) to substitute one good for the other. • If good 2 is measured in dollars, the MRS measures the consumer’s willingness to pay for an extra unit of good 1.

  4. Monotonicity: MRS negative (Strict) Convexity: MRS decreases as increases Assumptions on Preferences and the MRS

  5. Utility Function • Idea: assign a number to each consumption bundle, with higher numbers assigned to more-preferred bundles. • A utility function represents a preference relation :

  6. Utility Function: Does It Always Exist? • Q: Given a preference relation can we find a utility function that represents it? • A: If preferences are complete and transitive (plus a technical assumption called “continuity” is verified) we can. [Sufficient condition]

  7. Is Transitivity Necessary? • Q: Is it necessary that preferences are transitive for the existence of a utility function that represents them? • A: Yes. Otherwise:

  8. Utility is Just Ordinal E.g.: (1,1) (1,0.5) (0.4,1) (0.8,0.8) Can be represented in different ways:

  9. In General If : • 1) represents • 2) is a positive monotonic transformation Then, also represents

  10. Utility function: Indifference curves: or: Utility Functions and Indifference Curves

  11. Utility function: Indifference curves: or: Indifference Curves and Monotonic Transformations

  12. Utility function: Indifference curves: Perfect Substitutes

  13. Utility function: Perfect Complements

  14. Utility function: Indifference curves: Cobb-Douglas Preferences

  15. Cobb-Douglas • Most commonly used utility function in economics • Possible to assume without loss of generality that • Why?: apply to get

  16. Marginal Utility • Consider a consumer that is consuming • Q: By how much does his utility change as we increase by a very small amount his consumption of good 1? • A: Marginal utility of good 1:

  17. Marginal Utility and Units • The marginal utilities of good 1 and good 2 depend on the specific utility function we are using. • Consider: • Then:

  18. Marginal Utility and MRS • MRS only depends on preferences and not on their specific representation (utility function). • Marginal utilities can be used to compute the MRS between two goods.

  19. Computing the MRS • Consider an indifference curve: • Q: What is the slope of this indifference curve (MRS)? • A:

  20. The MRS and Monotonic Transformation • Consider a monotonic transformation • Q: What is the MRS in this case? • A:

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