A P.O. Is a C.E. © 1998 by Peter Berck

What Is It Good? §Sum of surplus and profits allows for policies that make income less evenly distributed. §A dam that increases banana plantation owners incomes by more than it decreases native American’s incomes increases surplus plus profits. §We can do better.

Pareto Optimal (Easy Definition) §An allocation is Pareto optimal if there is no way to make any agent better off without making some other agent worse off.

The First Welfare Theorem §A Competitive Equilibrium is a Pareto Optimum

First Welfare Theorem: Proof.

The setting §There are N goods, (x 1 …x N ) §There are N prices, (p 1 …p N ) §There are M consumers l each has an initial endowment or allocation of goods (a 1 …a N ) I §There are price taking profit maximizing firms.

Getting Simpler §Only two goods §Firm produces good two from good one §One consumer who begins life with an endowment (initial allocation) of good one l endowment = (a, 0)

The firm  Maximizes its profits,  (p 1,p 2 ), by its choice of input, x 1. l max p 2 x 2 - p 1 x 1 s.t. x 2 = F(x 1 ) l where F is the production function cost function p = mc get profits, supply curve, x 2 f (p 1,p 2 ) demand for factors x 1 f (p 1,p 2 ) from this problem. –f for firm

The consumer  income y = a p 1 +  §income = expenditure = p 2 x 2 + p 1 x 1 §Consumer maximizes happiness subject to income = expenditure l result is x 1 c (p 1,p 2 ) and x 2 c (p 1,p 2 ) consumers demand curves. Notice how income is folded in to the prices Notice that consumer started with a units of the first good, sold them, and repurchased x 1 c units

C.E. is Competitive equilibrium §A set of prices (p 1,p 2 ) at which §consumers max happiness or utility §producers max profits §and supply equals demand in all markets §a = x 1 c (p 1,p 2 ) + x 1 f (p 1,p 2 ) §x 2 f (p 1,p 2 ) =x 2 c (p 1,p 2 ) §the competitive prices are (p 1 *,p 2 *)

The Competitive Final Allocation §A single consumer’s final allocation is §x*=(x 1 c *, x 2 c *)= (x 1 c (p* 1,p* 2 ), x 2 c (p* 1,p* 2 ) ) l Where there are J consumers the final allocation is l A* = ((x 1 c (p* 1,p* 2 ), x 2 c (p* 1,p* 2 )) 1, l …,(x 1 c (p* 1,p* 2 ), x 2 c (p* 1,p* 2 ) ) J ) §The firms input output choice is l (x 1 f (p* 1,p* 2 ), x 2 f (p* 1,p* 2 ) ) =(x 1 f *, x 2 f *)

Pareto Optimal §A final allocation is Pareto preferred to another final allocation if no consumer is made worse off and at least one consumer is made better off. §An allocation is feasible if it is physically possible §When there is no feasible final allocation preferred to a given allocation, then the given allocation is Pareto Optimal

Proof of First Welfare Theorm §Plan l Posit a feasible better bundle x c ’ = (x 1 c ’,x 2 c ’) l show that x c ’ costs more than x c * (at prices p*) l show that profits are not more at x f ’ l use budget constraint to show that more of x 1 is demanded than exists l therefore x’ is not feasible l therefore x* is pareto optimal

Note §This is the real proof, save the one consumer and two good restriction. §This is Gerard Debreu’s proof. Berkeley Economics professor and Nobel Laureate. l (at least I learned it from his book Theory of Value)

Suppose §That there is a bundle x c ’ = (x 1 c ’,x 2 c ’) that is preferred by the consumer to the competitive equilibrium allocation x c * = (x 1 c *,x 2 c *) §That there is a firm input and output that makes this consumption possible, and that this input and output are technically possible: x 2 f ’ = F(x 1 f ’)

Likes better means costs more x1x1 x2x2 xc’xc’ xc*xc*

x’ costs more than x* §expenditure = p 2 x 2 c + p 1 x 1 c §p 2 * x 2 c ’ + p 1 * x 1 c ’ > p 2 * x 2 c * + p 1 * x 1 c *  income y = a p 1 * +  §p 2 * x 2 c ’ + p 1 * x 1 c ’ > p 2 * x 2 c * + p 1 * x 1 c *  =a p 1 * +   p 2 * x 2 c ’ + p 1 * x 1 c ’ > a p 1 * +   value of x’ is greater than consumers’ income

x’ doesn’t maximize profits at p* §(x 1 f*,x 2 f* ) gives the greatest value of profits of all those input-output combinations that satisfy the production function. l That was the definition of x f * ! §So if x f ’ isn’t equal to x f * then the profits at x f ’ must be less than at x f *   f *   f ’= x 2 f’ p 2 * -x 1 f’ p 1 *

putting it together §p 2 * x 2 c ’ + p 1 * x 1 c ’ > p 2 * x 2 c * + p 1 * x 1 c *  =a p 1 * +  l like better costs more   * >  ’= x 2 f’ p 2 * - x 1 f’ p 1 * l we already maximized profits at these prices  p 2 * x 2 c ’ + p 1 * x 1 c ’ >a p 1 * +  a p 1 * +  ’ §consumption has higher value than endowment plus profits

Can’t Be  P 2 * x 2 c ’ + p 1 * x 1 c ’ >a p 1 * +  a p 1 * +  ’ §P 2 * x 2 c ’ + p 1 * x 1 c ’ > a p 1 * - x 1 f’ p 1 * +x 2 f’ p 2 * § P 2 * (x 2 c ’-x 2 f’ ) + p 1 *(x 1 c ’+ x 1 f’ -a) > 0 §So the demand for at least one of the goods must exceed supply §So x’ isn’t feasible (more is bought than is sold or otherwise exists) §So x* is a pareto optima

The Point §When all goods are traded in a competitive market, the outcome is Pareto optimal. l This is the justification for “letting the market work” or non-interference by government. l It is not true when goods like “clean air services” are not traded freely. l It does not imply that income is distributed in an even fashion.

Does Ownership of the Air Matter? §The Coase theorem: it does not matter whether the polluter or pollutee owns the rights to clean air, the amount of clean air will be the same.

Two Firms Additional Profits From One More Unit of CAS Clean Air Services Firm 1Firm 2 No matter who owns the CAS to begin with, the firms always trade to the intersection of their bid curves.

Two Firms Additional Profits From One More Unit of CAS Clean Air Services Firm 1Firm 2 Dead weight loss is greatest, with no trade, if the Red firm is prohibited from polluting. If the red firm were liable for the damages of its pollution, it would pollute and pay the damages.

Smoker and Asthmatic §In this example there are two consumers, smoker and asthmatic. §There are two goods, clean air (CAS) and chocolate available in fixed supply. §The initial allocation of the goods makes a big difference for how clean the air is. §The initial allocation makes a big difference for how much each agent gets.

Edgeworth-Bowley Box §The width is the amount of CAS l Asthmatic uses CAS to breathe l Smoker, to smoke §The height is the amount of chocolate CAS Chocolate A S

Who Got What?

Picture of a P.O.

The “Lens” of Pareto Preferred

A Possible Trade to a P.O.

Which Are P.O.?

All Possible P.O.’S A S Does Endowment Matter Now???? Is there a Coase Theorem?

Coase Theorem (Revised) §When changes in income don’t change the demand for the environmental good. §And the costs of transaction between agents is small. §And the property rights in clean air services are defined. §Then, trading in clean air will lead to a unique amount of clean air and pollution.

The First Welfare Theorem §When all the goods are freely traded, a competitive equilibrium is a Pareto optimum. §There are many Pareto optima. l All of them can be achieved as competitive equilibrium starting at SOME initial allocation. §Agent’s have very different incomes in the different Pareto optima.

Gloria’s comment: §An economy can be perfectly (Pareto) efficient and perfectly disgusting.

Free Rider Problem §When I buy a candy bar, who benefits? §When I buy an SO 2 certificate, who benefits? l Do all the beneficiaries contribute to the cost of buying and retiring the certificate? l Nother example: 6 of us go out to dinner and split the bill. What happens to the size of our tab?

First Welfare Theorem and Pollution §The theorem applies to goods that are traded. §Consumers’ demand for clean air isn’t effective because: l Costs to much to trade small amounts of CAS. l Benefits from buying it do not accrue only to the consumer who bought it. §Government is needed to assure clean air.