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Microeconomics 3200/4200:Part 1

P. Piacquadio

p.g.piacquadio@econ.uio.no

October 8, 2014

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 1 / 21

Outline

1 WelfareIntroductionEfficiency in the sense of ParetoEquitySolutions

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 2 / 21

Where we are...

Firms (lectures 1 and 2) choose optimally how/what to produce: buyinputs and sell outputs to maximize profits.Consumers (lectures 3 and 4, ... and 5 for risk) choose optimally whatto consume: use resources and income to buy consumption goods.On the market (lecture 6), supply and demand meet: the pricemechanism (and a bunch of assumptions) guarantee that there is anequilibrium price and a competitive equilibrium.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 3 / 21

In this lecture

We will discuss the competitive equilibrium, and other allocations,from the perspective of “social welfare”.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 4 / 21

Social welfare

What is social welfare?I social welfare is a measure of collective well-being;I it is a measure of the social desirability of a situation.

How do we measure social welfare?I the social welfare function is a function (similar to individual utilities)

that ranks social alternatives (instead of individual consumptions);I unfortunately such functions are extremely controversial and difficult to

construct: A celebrated result by Arrow (1963) shows that there existsno such function satisfying very appealing properties;

I what we will do next is to focus on some of these properties.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 5 / 21

Outline

1 WelfareIntroductionEfficiency in the sense of ParetoEquitySolutions

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 6 / 21

Pareto efficiency

Pareto efficiency requires the social welfare to be increasing in thesatisfaction level of all individuals of the societyAn allocation a is Pareto superior to allocation a′ if:

I each agent is at least as good at a than at a′ andI some agent strictly prefer a to a′.

An allocation a is Pareto optimal if there is no feasible alternative a′

that is Pareto superior to a.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 7 / 21

The First Fundamental Theorem of Welfare Economics

Theorem. Assume preferences of all agents satisfy greed. Then ifa ≡ ([x∗] , [q∗] ,p) is a competitive equilibrium then the allocation([x∗] , [q∗]) is Pareto optimal.Some interpretations:

I “Markets are good:” they guarantee efficiency of the allocation;I “Adam Smith’s invisible hand” works;I without any public intervention, the price mechanism clears the excess

demands and gives the right incentives to firms and consumers toproduce and consume what is best.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 8 / 21

The Second Fundamental Theorem of Welfare Economics

Theorem. Assume preferences satisfy greed and convexity and thatproduction functions are convex. Then for each Pareto optimalallocation ([x∗] , [q∗]), there exists a price vector p 6= 0 and somewealth transfers across agents b = (b1, ...,bnh) such thata ≡ ([x∗] , [q∗] ,p) is a competitive equilibrium.Some interpretations:

I “Markets are great:” efficiency and redistribution can be separated;I make some lump-sum wealth transfers (if you need to) and let the

market operate;I efficiency and equity can be combined and both obtained through the

strength and role of the invisible hand.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 9 / 21

However...

... “the devil is in the detail”...these results are true when (and almost only when!) ALL ourassumptions are satisfied:

I price taking behavior; single price for each good;I greed and convexity of preferences; rationality of preferences;I convexity of technology; profit maximizing behavior;I perfectly divisible and privately appropriable goods.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 10 / 21

Furthermore...

... even if the competitive equilibrium was efficient, we would need tosolve the issue of the optimal quantity of wealth redistribution amongagents.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 11 / 21

Outline

1 WelfareIntroductionEfficiency in the sense of ParetoEquitySolutions

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 12 / 21

How to interpret equity?

There is no “obvious” definition of equity.In a setting without production, a possible interpretation is “equaldistribution of resources.”This is however too strong and, arguably not appealing, as agents arelegitimately different, have different preferences, and would like toconsume different things.A better interpretation (Steinhaus 1948) is equal-split lower bound:each agent should find its consumption at least as good as the equalsplit of the available resources.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 13 / 21

Envy freeness

A further alternative interpretation of equity is “envy freeness” (or also“fairness” or “permutation proofness”) by Foley (1967).An allocation a is envy free if each agent finds her/his assignment atleast as good as the assignment of anyone else; i.e. for each pair ofagents h,k = 1, ...,nh, Uh (xh)≥ Uh (xk).

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 14 / 21

The competitive equilibrium from equal split

Define the competitive equilibrium from equal split as follows.Each agent is assigned the consumption corresponding to acompetitive equilibrium where the wealth of each is equalized.Theorem. In a distribution economy, the competitive equilibrium fromequal split is efficient, envy free, and satisfies equal-split lower bound.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 15 / 21

Proof: intuition

Since it is a competitive equilibrium, it is efficient.By equal wealth and price uniqueness, everyone can choose in the sameset of alternatives: what is affordable for h is also affordable for k .Thus envy freeness is also satisfied: since everyone maximizes the ownutility over the same set of alternatives, each picks his/her favoriteand cannot be better-off with the one of someone else.Finally, equal-split lower bound follows from the equal-split ofresources being inside the set of affordable alternatives.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 16 / 21

Issues

When agents might be different with respect to non-transferablecharacteristics (such as skills, ability, handicaps, health, etc...) theremight be no allocation satisfying both efficiency and envy freeness.The society might not want to treat all equally. It is generally acceptedthat inequalities are bad, but... moderate inequalities can be tolerated.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 17 / 21

Outline

1 WelfareIntroductionEfficiency in the sense of ParetoEquitySolutions

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 18 / 21

Common solutions

Two solutions have been adopted by the literature to avoid thedifficulties of measuring social welfare, while combining equity andefficiency.1. What matters is wealth or income. Social welfare is measured as afunction of the wealth or income of each agent. Equality and (to someextent) inequality are easy to define.

I Examples are GDP, income inequalities, top income inequalities, etc...

2. What matters is a comparable measure of individual well-being.Social welfare is measured as a function of the well-being of eachagent. Equality and (to some extent) inequality are easy to define.

I Examples are utilitarianism, leximin social preferences, etc...

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 19 / 21

Alternative approaches

There are some in the literature: capability/functioning approach,equality of opportunity, resource-based approach, fair allocationtheory, social ordering functions,...If you are interested, I would invite you to attend the course on“Distributive Justice and Economic Inequality” in Spring 2015.

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 20 / 21

Thank you very much!

P. Piacquadio (p.g.piacquadio@econ.uio.no) Micro 3200/4200 October 8, 2014 21 / 21