Lesson 1. Walrasian Equilibrium in a pure Exchange Economy. General Model

Size: px

Start display at page:

Download "Lesson 1. Walrasian Equilibrium in a pure Exchange Economy. General Model"

Charleen Lewis
5 years ago
Views:

1 Lesson Warasian Equiibrium in a pure Exchange Economy. Genera Mode

2 Genera Mode: Economy with n agents and k goods. Goods. Concept of good: good or service competey specified phisicay, spaciay and timey. Assumption : There is a finite number k of goods. Assumption 2 : Goods can be consumed in any non-negative rea number. Goods are perfecty divisibe k The space of goods is R + Chapter 6 2

3 Genera Mode. Goods. A aocation of goods is a vector k x = ( x, x 2,..., x k ) R + Prices: There exists a market for each good and then a price. (there exist future markets for a future goods) Let p be the amount paid (in terms of a numeraire ) for each unit of good x, =,2,.k. p is + (good or scarce commodity), - (bas or bad commodity) 0 (free good) The price system is a vector: p=(p,p 2,,p,..,p k ) en R k Chapter 6 3

4 Genera Mode. Barter Economy The economy works out without the hep of the good money (or any other good as accounting measure) The mode is of perfect information (or perfect forecast) The mode is static. Steady state (the dynamic structure is not in the mode). Agents choose consumption pans for a their ives) Chapter 6 4

5 Genera Mode. Agents. Assumption. A finite number, n, of consumers. Consumer s aim= To choose a consumption pan according to her choice-criterion and her surviva (consumption set) and weath constraints. Choice criterion: An economic decision maker aways chooses her most preferred aternative from the set of avaiabe aternatives. Assumption 2 : consumers are price-takers. Description of an economic agent: Each consumer i defined by: Her consumption set X i i i i i k Her initia endowments W = ( w, w2,..., wk ) R+ Her preferences over baskets of goods. Chapter 6 5

6 Genera Mode. Agents. The consumption set: Exampe: Surviva. Assumption C: i k X R + i k X = R+ Assumption C2: X i is convex and cosed (perfect divisibiity) i i i i i k x = ( x, x2,..., xk ) X = R+ Chapter 6 6

7 Genera Mode. Preferences: Binary reationship between bundes of goods. f denotes strict preference: x f y means that x is stricty preferred (o is stricty better than) to y. denotes indiference: x y means that x and y equay preferred (is exacty as good as). < denotes weak preference x < y means that x is at east as preferred as (or is at east as good) as y. Chapter 6 7

8 Assumptions on preferences Assumption 2. Refexivity: x < x. Assumption 3. Transitivity: if x < y and y < z, then x < z. Assumption. Competeness: either x < y, or y < x, or both. Pre-order in X. The indiference reationship partitions X in equivaence cases, which are disjoints and exhaustives =Indiference sets with at east one eement (by refexivity) Chapter 6 8

9 Indiference set x 2 x x x,, x x y x x are equay preferred; x x x. x x Chapter 6 9

10 Genera Mode. Utiity function. Utiity function: Rue associating a rea number to each vector of goods in X: U: X R, such that x f y u(x) >u(y) ; x y u(x) =u(y). An isomorphism is need (an order preserving reationship betwen sets) between X y R, in order that preferences are represented by an utiity function. Let I be the set of equivaence casses. (I, f) is isomorphic to (Q, f), where Q is the set of rationa number, whenever set I is finite. If I is not finite, and additiona assumption is need to guarantee that preferences are representabe by utiity functions. Chapter 6 0

11 Genera Mode. Preferences Assumption 4. Continuity. For a x and y εx, the sets {x: x < y } and {x: x - y} are cosed. Theorem: If < satisfies competeness, refexivity, transitivity and continuty, then there exists an utiity function U: X R representing these preferences. U is continuos and satisfies that u(x) u(y) if and ony if x < y. U(.) is ordina and uniquey represents < if a the positive transformations of U(.) are incuded. If U(.) represents < and if f:r R is a monotone increasing function, then f(u(x)) aso represents < since f(u(x)) f(u(y)) if and ony if u(x) u(y). Chapter 6

12 Genera Mode. Preferences If we want U(.) to be increasing: Assumption 5. Strong Monotonicity (no-satiation): If x y and x y x f y( More is preferred to ess ) A weaker assumption is: Loca no-satiation: x εx ande>0, there exists an y ε X, such that x-y <e y fx. To guarantee we-behaved demand functions. Assumption 6. Stric convexity. Given x y and zεx, if x < z and y < z tx+(-t)y f z, for a t ε(0,). Assumption 6 : Weak convexity: If x < y, then tx+(-t)y < y, for a t ε(0,). Chapter 6 2

13 Sopes of Indifference Curves Good 2 Better Better Two goods a negativey soped indifference curve. Worse Worse Good Chapter 6 3

14 We-Behaved Preferences -- Convexity. x 2 x 2 +y 2 2 y 2 x x x +y y 2 z = x+y 2 y is stricty preferred to both x and y. Chapter 6 4

15 The weacky preferred set x 2 x WP(x), the set of bundes weaky preferred to x. WP(x) incudes I(x) I(x). x Chapter 6 5

16 Convex Preferences. Perfect Substitutes. U=x +x 2 x I 2 I Sopes are constant at -. 8 Bundes in I 2 a have a tota of 5 units and are stricty preferred to a bundes in I, which have a tota of ony 8 units in them. 5 x Chapter 6 6

17 Convex Preferences: Perfect Compements: U=Min{x,x 2 } x o I I 2 Since each of (5,5), (5,9) and (9,5) contains 5 pairs, each is ess preferred than the bunde (9,9) which contains 9 pairs. 5 9 x Chapter 6 7

18 Non-Convex Preferences x 2 z Better The mixture z is ess preferred than x or y. y 2 x y Chapter 6 8

19 More Non-Convex Preferences x 2 z The mixture z is ess preferred than x or y. Better y 2 x y Chapter 6 9

20 Sopes of Indifference Curves Good 2 Better Better Worse Worse One good and one bad a positivey soped indifference curve. Bad Chapter 6 20

21 Indifference Curves Exhibiting Satiation x 2 Better Better Better Satiation (biss) point Better x Chapter 6 2

22 Preferences and Utiity Theorem: If < satisfy the assumptions ()-(6), then < can be represented by a utiity function U(.), which is continuos, increasing and stricty quasi-concave. Consumer i is characterized by a) Her consumption set: X i b) Her preferences < i u i (.) c) Her initia endowments w i. Aocation: x=(x,x 2,.,x n ) a coection of n consumption pans. n n i i Feasibe aocation: x = w i= i= Chapter 6 22

23 Demand functions and aggregate demand. Existence and characterization. Question: Istherea pricevector p such that: ) each consumer i maximizes her u i (.) and 2) the n- consumption pans are compatibe?. Demand functions (Existence) Weiertrass Theorem: Let f be a rea continuous function defined in a compact set of an n-dimensiona space, then f achieves its extreme vaues (maximum and minimum) in some points of the set Impies: Let f: R n R be continuous and A R n a compact set, then there exits a vector x* soving: Max f(x) subject to xε A Chapter 6 23

24 Demand functions. Existence Consumer s probem: Max {x} u i (x i ) subject to k k i i px pw = = Notice:. If p»0, then the budget set is compact. 2. If u i is continuous and. is satisfied, then by Weiertrass Theorem there exists at east a x *i =x i (p, pw i ) maximizing the consumer s probem and which is continuous. If u i is stricty quasi-concave, then x *i =x i (p, pw i ) is unique. Demand function of agent i (vector) x *i =x i (p, pw i ). Then: If < is stricty convex, continuous and monotone, then the demand function exists (is we-defined) and is continuous in a its points. Chapter 6 24

25 Demand functions. Characterization. Max {x} u i (x i ) s.t. Associate Lagrangian: Λ Khun-Tucker Conditions. k k i i px pw = = i i i i i i i i i 2 k λ = 2 k λ = = ( x, x,..., x, ) u ( x, x,..., x ) [ p x p x ] For an interior soution: C.P.O.: i δ L δ u = λ p = 0, =,2...k i i δ x δ x k k L i i p x p w δλ = = δ = [ ] = 0 Chapter 6 25

26 Marshaian or ordinary demand functions. Net demand functions. Comparative Statics. Marshaian or ordinary demand functions of agent i (vector): x *i =x i (p, pw i ) Net demand functions of i (vector): x *i =x i (p, pw i )-w i Comparative Statics:. Effect of a change of a good initia endowment : Let pw i =M i. δx δx δ M δx δw δm δw δm *i i i = = i i p Chapter 6 26

27 Comparative Statics. Sutsky s equation. 2.Effect of a change in good s price: δ x δ x δ x δ M δ p δ p δm δ p * i i i = M + * i i i = M + * i i i i = * i x + δ x δ x δ x w δ p δ p δm δ x δh δ x δ x δ M δ p δ p δm δm δ p And rearranging the above terms: Modified Sutsky s equation: * i i i δx δh δx = x w δ p δ p δm * i i ( ) Chapter 6 27

28 Comparative Statics. Sutsky s equation: Norma good: Positive net demand Negative net demand Inferior good: Positive net demand Negative net demand δ δ δ δ δ δ * i i i x h x * i i = x w p p M δ x δ p δ δ * i x p * i * i δ x δ p δ δ * i x p < = = < 0?? 0 ( ) Chapter 6 28

29 Aggregate demand Addition set: With the individuaism hypotesis on preferences aggregate quantities=sum of individua quantities (there is not externaities) x *i =x i (p, pw i ) vector of i s demand funtions. X(p)= i x *i = i x i (p, pw i ), aggregate demand function of the Economy. At each market : x * i =x i (p, pw i ) is i s demand function of good, and X (p)= i x i (p, pw i ) is the market aggregate demand function of good. Chapter 6 29

30 Warasian Equiibrium: Let p=(p,p 2,,p k ) be a price vector. Each agent i: Max {x} u i (x i ) subjet to px i =pw i Soution: demand function of i: x *i =x i (p, pw i ) Aggregate demand: X(p)= i x *i = i x i (p, pw i ) Aggregate suppy: i w i. Is there a price vector p* such that i x i (p, pw i )= i w i, and with free goods i x i (p, pw i )6 i w i? Chapter 6 30

31 Warasian equiibrium: Let z(p)= i x i (p, pw i )- i w i be the excess demand function of the economy, and Let z j (p)= i x ji (p, pw i )- i w ji, be the excess demand function of good (market) j. A price vector p * 0, is a warasian equiibrium (or competitive equiibrium) if: z j (p * )=0, if j is scarce (p j * >0) z j (p * )<0 if j is a free good (p j * =0) Chapter 6 3

32 Exampes of Warasian equiibrium: p S p p* x(p) P* z(p) x 0 z(p) Chapter 6 32

33 Exampes of Warasian equiibrium: Free goods p x(p) S Z(P) p x 0 z(p) p*=0 z(p)<0 p*=0 Chapter 6 33

34 Exampes of Warasian equiibrium: Non existence case. p S p x(p) z(p) x 0 z(p) Chapter 6 34

35 Exampes of Warasian equiibrium: p S= x(p) p*=a p 0 p z(p)=0 x 0 z(p) Chapter 6 35

36 Warasian equiibrium: Properties of the excess demand function z(p). Let us come back to our mode: Is there a price vector p* such that a markets cear? First: properties of z(p). Notice: ) The budget set of each agent i does not vary if a prices are mutipied by the same constant: the budget set is homogeneous of degree zero in prices. 2) By ) The demand function is homogeneous of degree zero in prices: x i (kp,kpw i )= x i (p,pw i ). 3) The sum of homogeneous functions of degree r is another homogeneous function of degree r: the aggregate demand function is homogeneous of degree zero in prices: i x i (p,pw i ) is homogeneous of degree zero n prices. 4) The excess demand function z(p)= i x i (p,pw i )- i w i is homogeneous of degree zero in prices. 5) By the assumptions on < (convexity) the demand function is continuous and since the sum of continuous functions is continuous the aggregate demand function is continuous: z(p) is a continuous function. Z(p) is a continuous and homogeneous function of degree zero in prices. Chapter 6 36

37 Warasian equiibrium. Existence: Brower s fixed point Theorem. To show the existence of WE we wi make use a fixed point theorem. The proof of the existence is made by modeing the behavior of price-revision by a warasian auctionier unti equiibrium prices are reached. S p 0 z(p 0 ) S p z(p ) S auctionier agents S revises according to thesignofz(p) agents Transactions (exchanges) are ony made at equiibrium prices. Chapter 6 37

38 Warasian equiibrium. Existence: Brower s fixed point Theorem. Consider the mapping from a set into itsef: f: X X. Question: is there a point x such that x=f(x)? If this point exists it is caed a fixed-point. The warasian equiibrium is going to be defined as a fixed point of a mapping from the set of prices into itsef: S p* z(p*) S auctionier agents S does not revise prices if excess demands are zero or negatives (free goods) Chapter 6 38

39 Warasian equiibrium. Brower s fixed point Theorem. Brower s fixed point Theorem: Let S be a convex and compact (cosed and bounded) subset of some eucídean space and et f be a continuous function, f: S S, then there is at east an x in S such that f(x)= x. Notice: The Theorem is not about unicity. The Theorem gives sufficient conditions for the existence of fixed points. Chapter 6 39

40 Warasian equiibrium. Brower s fixed point Theorem. Examp: Let S=[0,] and et f: [0,] [0,] As S is a compact set, then if f is continuous there exists at east a point x such that f(x)=x f(x 4)= f(x 3 ) f(x) f(x) f(x) 0 x f(x 2 ) f(x ) f(x 0)= 0= x 0 x x 2 x 3 =x 4 Chapter 6 40

41 Warasian equiibrium. Brower s fixed point Theorem. How important are the assumptions?. f not continuous f(x) f(x) 0 0 There is no fixed point Two fixed points: one in x=0 and another in x= Chapter 6 4

42 Warasian equiibrium. Brower s fixed point Theorem. How important are the assumptions? 2. S not convex: S=S S 2 S S S 2 f(x) f(x) f(x) S Theres is no fixed point Two fixed points: one in x=0 and another in x= Chapter 6 42

43 Warasian equiibrium. Brower s fixed point Theorem. How important are the assumptions? 3. S not cosed 4. S not bounded f(x) f(x) 0 0 S={x:0<x6 } Fixed point x=0, but x is not in S S={x: x 0} No fixed point Chapter 6 43

44 Warasian equiibrium. Existence The WE existence proof is based on the appication of Brower fixed point Theorem to our probem To ook for set S (convex and compact).. The price set P (thesetofvectors(p,,p k ) with non-negative eements) is not compact: Is bounded from beow: p 0, for a =,,k, but not from above. It is a cosed set. Then P is not compact. 2. Normaize the set of prices P to make it a compact set: each absoute price p is substituted by a normaized p : p = k p j = p ' j ' with p =. For instance: p =4 and p 2 =6 p =4/0=0.4 and p 2 =6/0=0.6 and p +p 2 =. (reative prices) Chapter 6 44

45 Warasian equiibrium. Existence We consider the price vectors beonging to k- dimensiona unitary simpex. = : = k S p k 2 R+ p = p 2 Exampe: {(, 2) + : 2 } S p p R 2 p p = + = p Chapter 6 45

46 Warasian equiibrium. Existence Set S k- (the normaized set of prices) is: bounded: p 0 y p 6, for a =,,k cosed: {0,} beong to S k- convex: if p and p are in S k- (impying that p = and p =), then: p=λp +(-λ)p is in S k-, since p = λp + (-λ)p =λ p + (-λ) p = Chapter 6 46

47 Warasian equiibrium. Existence 3. As z(p) (and the demand functions) is homogeneous of degree zero in prices, prices can be normaized and demands can be expressed as functions of reative prices. z(p,p 2,,p k )= z(tp,tp 2,,tp k )= z(p,p 2,,p k ), with t=/ j p j. 4. A potentia probem: z(p) is continuous whenever prices are stricty positive. If some p j =0, by monotonicity of preferences, demands wi be infinite discontinuity z(p) coud be not we-defined in the simpex boundaries. Soution to the probem:modify the non-satiation (monotonicity) assumption: There exist satiation eves for a goods, but there is aways, at east, a good which is bought by the consumer and such that the consumer is never satiated of it. Chapter 6 47

48 Warasian equiibrium. Existence. Waras Law Waras Law ( identity ): For any p in S k-, it is satisfied that pz(p)=0, i.e., the vaue of the excess demand function is identicay equa to zero. Proof: pz(p)=p( i x i (p,pw i )- i w i )= i (px i (p,pw i )-pw i )= i 0=0. Consequences of Waras Law: a) If for a p»0, k- markets cear, then market k wi aso cear: p z +p 2 z 2 =0 by LW, p >0 y p 2 >0, if z =0, LW z 2 =0 b) Free goods: if p* is a WE and z j (p)<0, then p j *=0, in words if in a WE a good has excess suppy, then this good wi be free: p z +p 2 z 2 =0 by LW, if z 2 <0, LW p 2 =0. Chapter 6 48

49 Warasian equiibrium. Existence Theorem. Existence Theorem: If z: S k- R k is a continuous function and satisfies that pz(p)=0 (LW), then there is a p* in S k- such that z(p*)60. Proof: ) to show the existence of a fixed point p* and 2) to show that this fixed point p* is a WE. ) Define the mapping g: S k- S k- by pj + max(0, zj( p)) g j ( p) =, j =,2,..., k k max(0, z ( p)) + = Rue to revise prices. Note that g is a composed función : for any initia p in S k-, z(p) in R k is obtained and by the rue defining the mapping a new p in S k- is obtained. Chapter 6 49

50 Warasian equiibrium. Existence Theorem g(p) graphicay: S k- R k Sk- p z(p) p g(p) Chapter 6 50

51 Warasian equiibrium. Existence Theorem For instance: Suppose p =0.8 and z (p)=-2; p 2 =0.2 and z 2 (p)=8. Then: g (p)=[0.8+0]/[+0+8]=0.8/9=0.09 and g 2 (p)=[0.2+8]/[+8]=8.2/9=0.9. Notice: a) g is a continuous function, since z(p) is a continuous function and each function max(0,z j (p)) is continuous as we. b) g(p) is in S k- since p + p + max(0, z ( p)) max(0, z ( p)) k k j j j j j= j= gj( p) = j j max(0, z( p)) = = = = + + max(0, z( p)) k k The mapping g has an economic intuition: if z (p)>0, then p (as the warasian e subastador warasiano). Chapter 6 5

52 Warasian equiibrium. Existence Theorem As g is a continuous function mapping S k- into S k-, by Brower: there exists a p* such that p*=g(p*), that is: p + max(0, z ( p)) p = g ( p ) =, j =,2,... k max(0, ( )) * * * * j j j j k * + z p = 2) Now it has to be shown that vector p* is a WE (that z j (p*)60, j=,2,,k). From above: k * * * * j+ = j+ j = k * * * j max(0, ( )) = max(0, j( )) = p max(0, z( p )) p max(0, z ( p )) p z p z p Chapter 6 52

53 Warasian equiibrium. Existence Theorem Mutipying by z j (p*) and adding the k equations: k * * * * j( ) j max(0, ( )) = j( )max(0, j( * )) = k * * * * z p pjzj p = zj p zj * p = j j * * pz j j( p ) = 0 j z p p z p z p z p [ max(0, ( ))] ( ) ( )max(0, ( )) By Waras Law: and then: z p z p = j * * j( )max(0, j( )) 0 Each term of this sum is either zero or positive (since it is either 0 or z j (p*) 2 ). If z j (p*)>0 then the above equaity coud not be satisfied, then z j (p*)60 and p* is a WE. Chapter 6 53

u(x) s.t. px w x 0 Denote the solution to this problem by ˆx(p, x). In order to obtain ˆx we may simply solve the standard problem max x 0

u(x) s.t. px w x 0 Denote the solution to this problem by ˆx(p, x). In order to obtain ˆx we may simply solve the standard problem max x 0 Bocconi University PhD in Economics - Microeconomics I Prof M Messner Probem Set 4 - Soution Probem : If an individua has an endowment instead of a monetary income his weath depends on price eves In particuar,