General competitive equilibrium

General equilibrium: How does an idealized private ownership competitive economy works ?  l goods (indexed by j )  n individuals (households) (indexed by i )  k firms (or technologies) (indexed by f )  firm f ’s technology: a production set Y f   l that is closed, irreversible, convex and satisfies the possibility of inaction, impossibility of free production, and free disposal. u All goods are private (rival and excludable). u They are privately owned.   i j  0: quantity of good j initially owned by household i

General equilibrium: How does an idealized private ownership competitive economy works ? (2)  1   i f  0 : share of firm f owned by household i  Each firm is entirely owned (  i  i f = 1 for all f )  X i   l + : Consumption set of household i (convex and closed)   i : preferences of household i (reflexive, complete, transitive, continous, locally non- satiable and convex binary relation on X i ).

General equilibrium: How does an idealized private ownership competitive economy works ? (3)  An economy  = ( Y f, X i,  i,  i f,  i j ), i =1,…, n, j =1,…, l and f = 1,… k.  Economic problem: finding an allocation of the l goods accross the n individuals. u Some allocations are feasible, some are not. u A(  ): the set of all allocations of goods that are feasible for the economy .

General equilibrium: How does an idealized private ownership competitive economy works ? (4) u A(  ) is defined as follows: In words, A() is the set of bundles of goods that could be consumed in the economy given its technological possibilities, and the initial available resources (under the assumption that these resources are publicly owned)

A nice geometrical depiction of the set of feasible allocations: the Edgeworth box  Suppose Y f = {0 l } for all f (nothing is produced) u Then  is an exchange economy. u A(  ) in this case can be defined by: If l = n =2, we can represent the bundles that satisfy this weak inequality at equality on the following diagram

The Edgeworth Box Individual 1 individual 2 x22x22 x11x11 x12x12 x21x21 x  2 =   2 2  1 =   2 1

The Edgeworth Box Individual 1 individual 2 22 11 x

Pareto efficiency u Some feasible allocations of goods involve waste. u Some feasible allocations of goods do not exhaust the existing possibilities of mutual gains (called « win-win » situations in ordinary language) u Some feasible allocations of goods are not Pareto-efficient!

Pareto efficiency  Definition: an allocation x i j  A(  ) (for i = 1,…, n and j = 1,…, l ) is Pareto-efficient in A(  ) if, for any other allocation z i j  A(  ), having z h  x h for some individual h must imply that x g  z g i = 1,…, n. some individual g.  In words, an allocation x i j  A(  ) (for i = 1,…, n and j = 1,…, l ) is Pareto-efficient in A(  ) if it is impossible to find an allocation in A(  ) that everybody weakly prefers to x i j and that at least one person strictly prefers to x i j

Pareto-efficiency in an Edgeworth Box 1 2 x22x22 x11x11 x12x12 x21x21 x y 22 11 z

1 2 x22x22 x11x11 x12x12 x21x21 x y z 22 11 z is not Pareto- efficient

Pareto-efficiency in an Edgeworth Box 1 2 x22x22 x11x11 x12x12 x21x21 x y 22 11 Allocations in this zone are unanimously preferred to z z

Pareto-efficiency in an Edgeworth Box 1 2 x22x22 x11x11 x12x12 x21x21 x y 22 11 Allocation y (among other) is unanimously preferred to z z

Pareto-efficiency in an Edgeworth Box 1 2 x22x22 x11x11 x12x12 x21x21 x y 22 11 Allocation y is Pareto- efficient z

Pareto-efficiency in an Edgeworth Box 1 2 x22x22 x11x11 x12x12 x21x21 x y 22 11 So is x ! z

Pareto-efficiency in an Edgeworth Box 1 2 x22x22 x11x11 x12x12 x21x21 x y 22 11 So are all the allocations on the blue locus z

Pareto efficiency u A minimal normative requirement. u An inefficient allocation is unstatisfactory. u Yet Pareto-efficiency is hardly a sufficient requirement. u There are many Pareto-efficient allocations, and some of them may involve significant inequality u As Amartya Sen put it « a society may be Pareto-efficient and perfectly disgusting!

General Competitive equilibrium u What happens when all households and all firms take their decisions individually, taking as given a prevailing set of prices ? u Given prices, each firm chooses a production activity that maximizes its profits.  GIven prices, each household chooses a bundle of l goods that it most prefer. u Prices are such that these choices are mutually consistent (supply equal demand on all markets).

General Competitive equilibrium u Here is a formal definition.  A General Competitive Equilibrium (GCE) for the economy  = ( Y f, X i,  i,  i k,  i j ), i =1,…, n, j =1,…, l and f = 1,… k is a list ( p *, x i *, y f * ) with p *   l +, x i *  X i for i =1,…, n, y f *  Y f for f =1,… k such that: 

General Competitive equilibrium u Here is a formal definition.  A General Competitive Equilibrium (GCE) for the economy  = ( Y k, X i,  i,  i k,  i j ), i =1,…, n, j =1,…, l and f = 1,… k is a list ( p *, x i *, y f * ) with p *   l +, x i *  X i for i =1,…, n, y f *  Y f for f =1,… k such that:

General Competitive equilibrium u Here is a formal definition.  A General Competitive Equilibrium (GCE) for the economy  = ( Y k, X i,  i,  i k,  i j ), i =1,…, n, j =1,…, l and f = 1,… k is a list ( p *, x i *, y f * ) with p *   l +, x i *  X i for i =1,…, n, y f *  Y f for f =1,… k such that: and

General Competitive equilibrium u Here is a formal definition.  A General Competitive Equilibrium (GCE) for the economy  = ( Y f, X i,  i,  i k,  i j ), i =1,…, n, j =1,…, l and f = 1,… k is a list ( p *, x i *, y f * ) with p *   l +, x i *  X i for i =1,…, n, y f *  Y f for f =1,… k such that:

General Competitive equilibrium  Condition 1) says that given prices, and the budget constraint that these prices define (given initial endowments and ownerships of firms), household i chooses a bundle of goods that it most prefer in its budget set.  Condition 2 says that given prices firm k chooses a production activity in its production set that maximizes its profits. u Condition 3 says that choices made by firms and consumers are all mutually consistent (on every market, the demand for the good is never superior to the amount of good availabe (both as the result of production and initial endowments).

General-equilibrium in an Edgeworth Box (no production) 1 2 1212 1111  2121 2222

1 2 1212 1111  2121 2222

1 2 x22x22 x11x11 1212 1111  2121 2222 -p*1/p*2-p*1/p*2 (p * 1  p * 2  1 2 )/ p * 2

General-equilibrium in an Edgeworth Box (no production) 1 2 x22x22 x11x11 1212 1111  2121 2222 -p*1/p*2-p*1/p*2 (p * 1  p * 2  1 2 )/ p * 2

General-equilibrium in an Edgeworth Box (no production) 1 2 x22x22 x11x11 1212 1111  2121 2222 -p*1/p*2-p*1/p*2 (p * 1  p * 2  1 2 )/ p * 2 x 2* 1 x 1* 1

General-equilibrium in an Edgeworth Box (no production) 1 2 x22x22 x11x11 1212 1111  2121 2222 -p*1/p*2-p*1/p*2 (p * 1  p * 2  1 2 )/ p * 2 x 2* 1 x 1* 1 x 1* 2 x 2* 2

Excess demand correspondance  Condition 3 defines what is called the « excess demand correspondance » Z :  l +   l as follows: Where X Mi ( p ) is the Marshallian demand correspondance of household i at prices p and y f* ( p ) is the supply/demand correspondanceof firm f at prices p

Walras Law  Theorem: if consumers preferences satisfy local non-satiation, then, for all elements Z in the excess demand correspondance Proof: Under local non-satiation, one has for any household i, and any element x Mi of its Marshallian demand correspondance:

Walras Law  Theorem: if consumers preferences satisfy local non-satiation, then, for all elements Z in the excess demand correspondance Proof: summing this equality over all individuals yields (exploiting the fact that firms’ shares sum to 1)

A corrolary of Walras Law  Theorem: if an element Z of the excess demand correspondance satisfies Walras law and if Z j ( p )  0 for all goods j, then p g = 0 for any good g for which Z g (p) < 0. Proof: obvious Interpretation: if the market for a good is in strict excess supply at a CGE, then the price of this good must be zero (example: sand, stones, etc.)

Consequence of this corrolary  If ( p *, x i *, y k * ) is CGE for an economy, then for every good g for which p* g > 0 one must have Z g (p*) = 0. u We are going to use this to prove the existence of CGE for any economy satisfying our assumptions. u Establishing the existence of CGE has been one of the major achievement of the 20th century mathematical economics (finalized in fifeties through the work of Arrow and Debreu. u Argument is based on the fact that certain functions admit Fixed Points.

Fixed points u Many existence theorems in mathematical economics and game theory (Nash equilibrium, GCE, etc.) are consequences of mathematical theorems known as « fixed point » theorems. u Two such theorems are particularly useful: Brouwer’s fixed point theorem (that deals with functions) and Kakutani’s fixed point theorem that deals with correspondances. u In order to understand these theorems, we must first understand what is a fixed point.  Definition: Given a function f : A  A, we say of an element a  A that it is fixed point of f if f(a) = a u Some functions do not admit fixed points.  For example, the function f that assigns to every alive individual whose father is alive this father does not have fixed point (because nobody is his or her own father)

Brouwer’s fixed point theorem u The mathematician Brouwer established (in 1912) a theorem guaranteeing that a (real-valued) function admits a fixed point.  Brouwer’s fixed point Theorem: Let A be a convex and compact subset of  k and let f : A  A be a continuous function. Then, there exists an element a  A that is a fixed point of f  Let us illustrate the theorem when A is a convex and compact subset (and therefore an interval) of .

Brouwer fixed point theorem A A x y y = x f

Brouwer fixed point theorem A A x y y = x Fixed points f

Brouwer fixed point Theorem A A x y f The assumptions of Brouwer’s theorem are all important

Brouwer fixed point theorem A A x y f For example the function f does not have any fixed point

Brouwer fixed point theorem A A x y f but f is not a function from A to A

Brouwer fixed point theorem A A x y f This function f (that is not continuous) does not have fixed point either!

Brouwer fixed point theorem A A x y f Similarly, if A is open above a continuous function like f does not have fixed points

Fixed points u Brouwer’s fixed point theorem applies to functions. Yet the CGE can be viewed as the fixed point of a correspondance of price adjustment (based on excess demand). u Kakutani (1941) has generalized Brouwer’s fixed point theorem to correspondances.  Kakutani fixed point theorem: Let A be a convex and compact subset of  l and let C : A  A be correspondance upper-semi continuous. Then, there exists an element a*  A that is a fixed point of C (and that is therefore such that a*  C ( a *)). u John Nash in 1950 has used this theorem to show the existence of a Nash equilibrium in non-cooperative games. u Debreu (1959) has used it as well in his proof of the existence of CGE.

Upper Semi-continuity ? A A x y C The correspondence C is upper semi-continuous

Upper semi-continuity ? A A x y C The correspondence C is upper semi-continuous

Upper semi-continuity ? A A x y C The correspondence C is not Upper semi-continuous

Berge (1959) Maximum Theorem  Theorem: Let A and B be two subsets of  k with A compact and let  : A  B   be a continuous function. Then, the correspondence C : B  A defined by: is upper semi continuous

Applies Berge Maximum Theorem (1) u To the profit maximization program. Here, set A of the theorem is the production set Y k Problem: Y f is not compact (it is closed but possibly unbounded). Solution: Maximize profits on the set Y f  {y   l : y  -  } \  l - - (exclude plans that use more inputs than what is initially available and/or that uses input without producing any output).

Applies Berge Maximum Theorem (2) u To the consumer’s utility maximization program: Here, set A of the theorem is the budget set (that is compact)

Existence of a GCE (Arrow-Debreu) (1)  An economy  = ( Y f, X i,  i,  i f,  i j ), i=,…, n, j =1,…, l and f = 1,… k admits a GCE if:  Y f satisfies closedness, irreversibility, impossibility of free production, possibility of inaction, free disposal and convexity fo r f = 1,…k  X i is a closed and convex subset of  l + containing a bundle x i distinct from (  i 1, …,  i l ) such that (  i 1, …,  i l ) > x i for i =,…, n   i is reflexive, transitive, complete, continuous, locally non-satiable and convex for i =,…, n.

Existence of a GCE (Arrow-Debreu) (2) u Proof: We find convenient to limit our quest for equilibrium prices to the set: S l-1 = {( p 1,…, p l )  l + : p 1 +…+ p l = 1} u This does not involve any loss of generality since agents’ behavior at given prices is unaffected by proportional changes in all prices u Hence, facing prices ( p 1,…, p l ) is equivalent to facing prices ( p 1 / ( p 1 +…+ p l ),…, p l / ( p 1 +…+ p l )) u S l-1 is clearly compact (it is called a « simplex ») u Define the correspondance A : S l-1  S l-1 as follows:

Existence of a GCE (Arrow-Debreu) (3)

u The set S l-1 is compact u The correspondance A is upper semi- continuous (as is the excess demand correspondance by Berge Maximum theorem) u Hence the correspondance A admits a fixed point ( p * 1,…, p * l ) u This fixed point of A is a GCE of the underlying economy. u Details: see Debreu (1959). Existence of a GCE (Arrow-Debreu) (3)

1 st Welfare theorem  If ( p *, x i *, y f * ) is a GCE for the economy  = ( Y f, X i,  i,  i k,  i j ), i =1,…, n, j =1,…, l and f = 1,… k, and if  i is reflexive, complete, transitive, and locally non-satiable, then the allocation x i *, (for i =1,…, n ) is Pareto-efficient in A(  ).  Proof: by contradiction, suppose that ( p *, x i *, y f * ) is a GCE for the economy  = ( Y f, X i,  i,  i k,  i j ), i =1,…, n, j =1,…, l and f = 1,… k, but assume that x i *, is not Pareto-efficient in A(  ).

1 st Welfare theorem  If ( p *, x i *, y f * ) is a GCE for the economy  = ( Y f, X i,  i,  i k,  i j ), i =1,…, n, j =1,…, l and f = 1,… k, and if  i is reflexive, complete, transitive, and locally non-satiable, then the allocation x i *, (for i =1,…, n ) is Pareto-efficient in A(  ).  Proof: by contradiction, suppose that ( p *, x i *, y f * ) is a GCE for the economy  = ( Y f, X i,  i,  i k,  i j ), i =1,…, n, j =1,…, l and f = 1,… k, but assume that x i *, is not Pareto-efficient in A(  ). Hence, there exists an allocation x i (for i =1,…, n ) in A(  ) such that:

1 st Welfare theorem  For at least one household h (1) Since x i  A (), there exists productie activities y f  Y f (for f =1,… k ) such that: (2) Condition (1) implies that, for all household i, one has:

1 st welfare theorem  And, for at least one household h, one has: Summing inequalities (3) over all households (with (4) holding for at least one of them) yields: (3) (4)

1 st welfare theorem  since, for all f Moreover, since y f * maximises firm f profits at prices p *, one has, for any such firm: (5) (6)

1 st welfare theorem  Substituting (6) into (5), one gets: Which contradicts the requirement that Inequality (2) holds for every good j. QED.

Meaning of the theorem u If all goods that matter for humans and all technologies that enable the production of certain goods out of others are privately owned, then the free working of the market – provided that it operates under perfect competition – leads to a Pareto efficient resources allocation.

This theorems points naturally to the limitation of the competitive market mechanism u Some important goods should not or can not be privately appropriate (non-rivalry or impossibility of exclusion). u Some markets can ot be perfectly competitive because the efficient scale of production is large relative to the market) u Some markets (insurance among others) can not appear because of severe problems of information asymetries (moral hazard and/or adverse selection)

Limits of this theorem u Efficiency is not everything! u There are many efficient allocations. u One can be efficient while being « unjust » (this requires of course a definition of « justice ») u The 2 nd welfare theorem answers in part to these limitations. u 2 nd welfare theorem: any Pareto-efficient allocation can be obtained as the result of the integrated funtionning of competitive markets provided that lump sum redistribution of the purchasing power among households be initially performed.

2 nd wefare theorem  If  = ( Y f, X i,  i,  i k,  i j ), pour i =1,…, n, j =1,…, l and f = 1,… k is an economy satisying all assumptions that guarantee the existence of a CGE and if the allocation ( x i *) i =1,…, n is Pareto efficient in A(  ), then there are (possibly negative) lump sum taxes T i for i =1,…, n, satisfying T 1 +…+ T n = 0, prices p 1 *,…, p l *   + l and productive activities y f *  Y f (for f = 1,… k ) such that: 

2 nd wefare theorem  If  = ( Y f, X i,  i,  i k,  i j ), pour i =1,…, n, j =1,…, l and f = 1,… k is an economy satisying all assumptions that guarantee the existence of a CGE and if the allocation ( x i *) i =1,…, n is Pareto efficient in A(  ), then there are (possibly negative) lump sum taxes T i for i =1,…, n, satisfying T 1 +…+ T n = 0, prices p 1 *,…, p l *   + l and productive activities y f *  Y f (for f = 1,… k ) such that: 

2 nd wefare theorem  If  = ( Y f, X i,  i,  i k,  i j ), pour i =1,…, n, j =1,…, l and f = 1,… k is an economy satisying all assumptions that guarantee the existence of a CGE and if the allocation ( x i *) i =1,…, n is Pareto efficient in A(  ), then there are (possibly negative) lump sum taxes T i for i =1,…, n, satisfying T 1 +…+ T n = 0, prices p 1 *,…, p l *   + l and productive activities y f *  Y f (for f = 1,… k ) such that: 

1 2 x22x22 x11x11 1212 1111  2121 2222 -pe1/pe2-pe1/pe2 (p e 1  p e 2  1 2 )/p e 2 2 nd welfare theorem in an Edgeworth Box (p e 1  2 1 +p e 2  2 2 )/p e 2 x 1* 1 x 2* 1 -p*1/p*2-p*1/p*2 (p * 1  p * 2  T 2 )/p * 2 (p * 1  p * 2  T 1 )/p * 2

The 2 nd welfare theorem rides on much more assumptions than the first u For example, it may not hold if preferences are not convex. u Let us illustrate this graphically.

1 2 x22x22 x11x11 1212 1111 2121 2222 The 2 nd welfare theorem requires convexity of preferences  (p * 1  p * 2  T 1 )/p * 2 -pe1/pe2-pe1/pe2 (p * 1  p * 2  T 2 )/p * 2