Problem Set 1: Consumer Theory

1. Problem Set 1: Consumer Theory Microeconomics 2 Matthew Robson University of York

2. Question 1 Assuming that the weak preference relation is transitive, show that: where x, y and z are consumption bundles.

3. Question 1 Weak Preference Relation: xRy - x is at least as good as y Strict Preference Relation: xPy- x is better than y. Indifference Relation: xIy - x and y are equally good. Assumptions: • R is reflexive, i.e. xRx • R is transitive, i.e. if xRy and yRz, then xRz • R is connected (or complete), i.e. xRy or yRx

4. Question 1 Therefore, by transitivity of R. Need to show that the opposite is not true, . Suppose was true. Then by transitivity of R, which contradicts . Thus, we have:

5. Question 2 Carefully illustrate in diagrams indifference curves for the following utility functions:

6. Question 2i) Perfect Substitutes

7. Question 2ii) Perfect Complements If If

8. Question 2iii)

9. Question 3 Suppose the consumer’s utility function is , where , she has income and faces prices and for commodities 1 and 2, respectively. Derive the consumer’s Marshallian demand functions.

10. Question 3 Lagrangian: FOC’s: (1) (2) (3)

11. Question 3 Using (1) & (2): Plug into (3):

12. Question 4 For the utility function , derive the corresponding indirect utility function and the expenditure function.

13. Question 4 Indirect Utility Function: (1)

14. Question 4 Dual Problem – choose and to minimise expenditure, subject to a utility constraint. FOC’s: (1) (2) (3)

15. Question 4 Using (1) & (2): Plug into (3): (4) (5)

16. Question 4 Expenditure Function: (6)

17. Question 4 Due to duality, the Indirect Utility Function and Expenditure Function are inverses of one another: Therefore: Rearrange to get: Or: (7)

18. Question 4 Each method leads to slightly different equations… But they should be equivalent.