Advanced Microeconomics . 1 6

1. Advanced Microeconomics .16 Mikołaj Czajkowski

2. Overview • Review • Pareto optimality • Positive representative consumer • Normative representative consumer • Utility possibility set • Pareto frontier of the utility possibility set • Linear social welfare function • Social welfare maximization problem

3. Pareto optimality • A feasible allocation is Pareto optimal(Pareto efficient) if there is no other allocation that Pareto dominates it • There is no feasible allocation such that: • No concern for distributional aspects

4. A positive representative consumer • When can we treat the aggregate demand function as if it was generated by a fictional representative consumer? • A positive representative consumer exists if there is a rational preference relation such that the aggregate demand function is the Walrasian demand function generated by this preference relation • Fictional individual whose utility maximization problem when facing society’s budget set would generate the economy’s aggregate demand function

5. A normative representative consumer • When can we treat the aggregate demand function as if it was generated by a fictional representative consumer and use his preferences as a measure of aggregate social welfare? • Normative representative consumer • Necessary condition – positive representative consumer exists • A (Bergson-Samuelson) social welfare function is a function that assigns a utility value to each possible vector of utility levels for the consumers in the economy • Function that expresses society’s judgements on how different distributions of individual utilities are compared

6. A normative representative consumer • Using the social welfare function we may try to: • (redistribute wealth to maximize social welfare) • Value of this function would define social indirect utility function • The positive representative consumer is a normative representative consumer relative to social welfare function if so defined social indirect utility function exists • Positive vs. normative representative consumerpossible that the former exists and the latter not

7. A normative representative consumer • If there is a normative representative consumer • His preferences have welfare significance • The aggregate demand function can be used to make welfare judgements • Always assuming optimally distributed level of wealth!

8. Utility possibility set • – a family of (continuous) utility functions representing the preference relations of consumers • Attainable vectors of utility levels for the economy specified by is the utility possibility set:

9. Utility possibility set • For two-consumer economy: • From the definition of Pareto optimality, the utility values of Pareto optimal allocation must belong to the boundary of the utility possibility set • Does the reverse hold? • Assume – the set is closed

10. Pareto frontier utility possibility • Pareto frontier utility possibility: • It follows that: • A feasible allocation is Pareto optimum if and only if

11. Pareto frontier utility possibility • Proof: If , then there is such that for all :and for some : . But only if there is a feasible allocation such that for all . Hence Pareto dominates . Conversely, if is not a Pareto optimum, then it is Pareto dominated by some feasible , which means that for all and for some . Hence .

12. Pareto frontier utility possibility • If every and every is convex, and if the utility functions are concave, then the utility possibility set is convex.

13. Linear social welfare function • Assume society’s distributional principles can be summarized in a social welfare functionassigning social utility values to the various possible vectors of utilities for the consumers, and is linear: • For we have • because social welfare nondecreasing in the consumer’s utility levels

14. Social welfare maximization problem • Which points in the utility possibility set maximize our measure of social welfare?

15. Social welfare maximization problem • If is a solution to SWMP with , then • (i.e. is the utility vector of a Pareto optimal allocation) • Every linear social welfare optimum with weights is Pareto optimal • If is convex, then for any , there is , , such that for all : • (i.e. is a solution to the SWMP) • Every Pareto optimal allocation (and hence, every Walrasian equilibrium) is a social welfare optimum for some welfare weights • If is not convex is every Pareto optimum still a maximum of SWMP?

16. Social welfare maximization problem • Using the social welfare weights associated with a particular Pareto optimal allocation (e.g. Walrasian equilibrium) is equivalent to the case of the welfare optimum in a single-consumer, single-firm economy • Utility function of a normative consumer: • Let be the aggregate production set • Pair is a solution to the problem:

17. Social welfare maximization problem

18. Literature • Readings: • MC: 16E • V:17.9