General competitive equilibrium

1. General competitive equilibrium

2. 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 k) • firm k’s technology: a production set Yk l that is closed, irreversible, convex and satisfies the possibility of inaction, impossibility of free production, and free disposal. • All goods are private (rival and excludable). • They are privately owned. • ij 0: quantity of goodjinitially owned by householdi

3. General equilibrium: How does an idealized private ownership competitive economy works ? (2) • 1  ik 0 : share of firm k owned by household i • Each firm is entirely owned (iik = 1for all k) • Xi 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 Xi).

4. General equilibrium: How does an idealized private ownership competitive economy works ? (3) • An economy  = (Yk,Xi,i,ik,ij), i =1,…,n, j =1,…,landk = 1,…K. • Economic problem: finding an allocation of the l goods accross the n individuals. • Some allocations are feasible, some are not. • A(): the set of all allocations of goods that are feasible for the economy .

5. General equilibrium: How does an idealized private ownership competitive economy works ? (4) • 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)

6. A nice geometrical depiction of the set of feasible allocations: the Edgeworth box • Suppose Yk = {0l} for all k (nothing is produced) • Then  is an exchange economy. • 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

7. The Edgeworth Box x21 individual 2 2 = 12 + 22 x x22 x12 Individual 1 x11 1= 11 + 21

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

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

10. Pareto efficiency • Definition: an allocation xij A() (for i = 1,…,n and j = 1,…,l) is Pareto-efficient in A() if, for any other allocation zij A(), having zhxh for some individual h must imply that xgzgi = 1,…,n. some individual g. • In words, an allocation xij 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 xijand that at least one person strictly prefers to xij

11. Pareto-efficiency in an Edgeworth Box x12 2 x21 y 2 x z x11 1 1 x22

12. Pareto-efficiency in an Edgeworth Box x12 2 x21 y 2 z is not Pareto- efficient x z x11 1 1 x22

13. Pareto-efficiency in an Edgeworth Box x12 2 x21 y 2 Allocations in this zone are unanimously preferred to z x z x11 1 1 x22

14. Pareto-efficiency in an Edgeworth Box x12 2 x21 y 2 Allocation y (among other) is unanimously preferred to z x z x11 1 1 x22

15. Pareto-efficiency in an Edgeworth Box x12 2 x21 y 2 Allocationy is Pareto- efficient x z x11 1 1 x22

16. Pareto-efficiency in an Edgeworth Box x12 2 x21 y 2 So isx! x z x11 1 1 x22

17. Pareto-efficiency in an Edgeworth Box x12 2 x21 y 2 So are all the allocations on the blue locus x z x11 1 1 x22

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

19. General Competitive equilibrium • What happens when all households and all firms take their decisions individually, taking as given a prevailing set of prices ? • 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. • Prices are such that these choices are mutually consistent (supply equal demand on all markets).

20. General Competitive equilibrium • Here is a formal definition. • A General Competitive Equilibrium (GCE) for the economy  = (Yk,Xi,i,ik,ij), i =1,…,n, j =1,…,landk = 1,…K is a list (p*,xi*,yk*) with p* l+, xi* Xi for i =1,…,n,yk* Yk for k =1,…K such that: 

21. General Competitive equilibrium • Here is a formal definition. • A General Competitive Equilibrium (GCE) for the economy  = (Yk,Xi,i,ik,ij), i =1,…,n, j =1,…,landk = 1,…K is a list (p*,xi*,yk*) with p* l+, xi* Xi for i =1,…,n,yk* Yk for k =1,…K such that:

22. General Competitive equilibrium • Here is a formal definition. • A General Competitive Equilibrium (GCE) for the economy  = (Yk,Xi,i,ik,ij), i =1,…,n, j =1,…,landk = 1,…K is a list (p*,xi*,yk*) with p* l+, xi* Xi for i =1,…,n,yk* Yk for k =1,…K such that: and

23. General Competitive equilibrium • Here is a formal definition. • A General Competitive Equilibrium (GCE) for the economy  = (Yk,Xi,i,ik,ij), i =1,…,n, j =1,…,landk = 1,…K is a list (p*,xi*,yk*) with p* l+, xi* Xi for i =1,…,n,yk* Yk for k =1,…K such that:

24. 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. • 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). prices, each household chooses a bundle of l goods that it most prefer.

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

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

27. General-equilibrium in an Edgeworth Box (no production) 2 21 (p*111+ p*212)/p*2 -p*1/p*2 22 12  x11 1 11 x22

28. General-equilibrium in an Edgeworth Box (no production) 2 21 (p*111+ p*212)/p*2 -p*1/p*2 22 12  x11 1 11 x22

29. General-equilibrium in an Edgeworth Box (no production) 2 21 x2*1 (p*111+ p*212)/p*2 -p*1/p*2 22 12  x11 1 x1*1 11 x22

30. General-equilibrium in an Edgeworth Box (no production) 2 21 x2*1 (p*111+ p*212)/p*2 -p*1/p*2 x1*2 x2*2 22 12  x11 1 x1*1 11 x22

31. Excess demand correspondance • Condition 3 defines what is called the « excess demand correspondance » Z: l+ l. as follows: Where xjMi(p) is the Marshallian demand of good j by household i at prices p and yjk*(p) is the net supply of good jby the firm k at prices p

32. Walras Law • Theorem: if consumers preferences satisfy local non-satiation, then: Proof: see blackboard

33. A corrolary of Walras Law • Theorem: if an excess demand correspondance Z satisfies Walras law and if Zj(p)  0 for all goods j, then pg = 0 for any good g for which Zg(p) < 0. Proof: obvious (see blackboard) 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.)

34. Consequence of this corrolary • If (p*,xi*,yk*)is CGE for an economy, then for every good g for which pg > 0one must have Zg(p) = 0. • We are going to use this to prove the existence of CGE for any economy satisfying our assumptions. • 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. • Argument is based on the fact that certain functions admit Fixed Points.

35. Fixed points • Many existence theorems in mathematical economics and game theory (Nash equilibrium, GCE, etc.) are consequences of mathematical theorems known as « fixed point » theorems. • 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. • 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 • Some functions do not admit fixed points. • For example, the function fthat assigns to every alive individual whose father is alive this father does not have fixed point (because nobody is his or her own father)

36. Brouwer’s fixed point theorem • The mathematician Brouwer a established (in 1912) a theorem guaranteeing that a (real-valued) function admits a fixed point. • Brouwer’s fixed point Theorem: Let Abe a subset convex and compact 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 Ais a convex and compact subset (and therefore an interval) of .

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

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

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

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

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

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

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

44. Fixed points • 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). • Kakutani (1941) has generalized Brouwer’s fixed point theorem to correspondances. • Kakutani fixed point theorem: Let Abe 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 thata* C(a*)). • John Nash in 1950 has used this theorem to show the existence of a Nash equilibrium in non-cooperative games. • Debreu (1959) has used it as well in his proof of the existence of CGE.

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

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

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

48. Berge (1959) Maximum Theorem • Theorem: Let A and Bbe two subsets of k with Acompact and let :AB beacontinuous function. Then, the correspondence C: B Adefined by: is upper semi continuous

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

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