Microeconomics, Block I Part 2

Transcription

1 Microeconomics, Block I Part 2 Piero Gottardi EUI Sept. 20, 2015 Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

2 Pure Exchange Economy H consumers with: preferences described by U h : R L +! R and resources ω h 2 R L +, h = 1,.., H (income comes from the sale of individual endowment of resources). Economic Problem/Activity: (re-)allocate resources among the consumers allocation: array x = (x 1,.., x H ) 2 H R L + feasible if it is consistent with available resources: h x h h ω h In the special case L = 2, H = 2, feasible allocations can be graphically represented using the Edgeworth Box. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

3 Pareto E ciency Evaluate alternative allocations using agents preferences: Pareto E ciency (PE): An allocation x 2 R L + H is Pareto e cient if: (i) it is feasible (ii) there is no other feasible allocation ^x 2 R L + H which allows to improve agents welfare (weakly for all, strictly for at least one): U h (ˆx h ) U h (x h ) for all h, U h0 (ˆx h0 ) > U h0 (x h0 ) for at least some h 0 When the above inequalities hold, we say ^x Pareto dominates x. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

4 Pareto E ciency cts. Note: Pareto dominance de nes a strict (social) preference relation over the set of allocations R L + H. A weak social preference relation is analogously de ned; it is however incomplete [why?]. Hence, a solution of the social (multi-agent) choice problem of nding a feasible allocation that weakly Pareto dominates all other feasible allocations cannot generally be found. On the other hand, there are typically (many) feasible allocations that are not Pareto dominated by some other feasible allocation: Pareto e ciency is still a useful criterion to evaluate allocations. + Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

5 How to nd PE allocations PE allocations are solutions of the problem: for some h and some given max x U h (x h ) U h 0 (x s.t. h0 ) Ū h0 for all h 0 6= h (1) h x h h ω h 0 Ūh. h06=h Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

6 How to nd PE allocations cts. FOCs (for an interior solution, under A.1 0 and di erentiability of U h for all h): DU h = ρ µ h0 DU h0 = ρ for all h 0 6= h x h = ω h h h where µ h0 and ρ are the Lagrange multipliers of the h 0 6=h two sets of constraints in (1). Under A.2 FOCs are also su cient conditions for a Pareto optimum. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

7 How to nd PE allocations cts. Thus: DU h = µ h0 DU h0 for all h 0 6= h utility gradients are collinear for all agents (or: MRS are equalized across agents) Varying the values of Ū h0 for h 0 6= h the solution typically changes ) Obtain set of all Pareto e cient allocations. Finding PE allocations in the Edgeworth box. Examples:.. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

8 Utility Possibility set and Welfare weights Utility possibility set: image of feasible allocations in the space of utility levels ( ) UP = U h H = U h (x h H ) H h=1 R+ ; for x 2 L, x h ω h h=1 h h PE allocations support points on the outer (NorthEast) boundary of UP: U h H 2 UP : Ûh H h=1 2 UP s.t. Û h U h, > for some h Under what conditions can we say UP is closed? and convex? and has a downward sloping outer boundary? [see HW] Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

9 Utility Possibility set and Welfare weights cts. PE allocations can also be found (under a mild extra condition) as solutions of the following problem of maximizing social welfare: max x h ξ h U h (x h ) s.t. h x h h ω h for any given set of strictly positive (welfare) weights ξ h H equivalent to nding a point on the outer boundary of UP h=1. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

10 Competitive Markets Various mechanisms are possible to attain a reallocation of existing resources: bargaining, voting, dictatorship, markets,... Will focus on the latter: allocations attained when agents trade in perfectly competitive markets. p such that x =.., x h (p, p ω h ),.. is a feasible allocation (note: m h = p ω h ). Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

11 Competitive Equilibrium More formally: Competitive Equilibrium: is a price vector p and an allocation x =.., x h,.. such that: (i) for all h, x h = x h (p, p ω h ), and (ii) x is feasible. That is: consumers optimize and markets clear. Finding CE allocations in the Edgeworth box: draw the o er curves (consumption bundles demanded at some price) for each consumer. [properties?] CE obtains when o er curves intersect (except at endowment point) Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

12 Competitive Equilibrium in Production Economies representative rm with technology Y R L Allocation: (y, x) 2 Y R L + H feasible if: h x h h ω h + y Pareto e cient: def. unchanged (need only to replace feasibility notion on p.2 with the one above) FOCs for Pareto e ciency: must add the following condition (when Y is representable by a di erentiable production function, with commodity 1 as output and all other goods as inputs: y 1 = f (z 2,.., z L )): ρ 1 f z j = ρ j for j = 2,..L Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

13 Competitive Equilibrium in Production Economies Competitive Equilibrium with Production: is a price vector p and an allocation ȳ, x such that: (i) for all h, x h = x h (p, p ω h + θ h π), (ii) ȳ 2 arg max y 2Y p y, and π = p ȳ (iii) ȳ, x is feasible. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

14 Welfare Properties: FWT Trade is voluntary, hence equilibrium allocations will satisfy individual rationality: U h (x h ) U h (ω h ) for all h, and more: Proof. First Welfare Theorem: Under the Assumptions..., all competitive equilibrium allocations are Pareto e cient Suppose x =.., x h,.. is a competitive equilibrium allocation for some price p and is not Pareto e cient. Then there must be another allocation ^x =.., ˆx h,.. which is feasible and Pareto improves upon x. But since x h is the optimal choice of consumer h at prices p, under... U h (ˆx h ) > U h (x h ) implies p ˆx h > p ω h for all h and also, under..., U h (ˆx h ) = U h (x h ) implies p ˆx h p ω h, with all the previous inequalities being strict for at least some h. Summing the latter inequality over h yields p h ˆx h > p h ω h which contradicts the feasibility of ^x (since, under..., p 0). Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

15 FWT cts. Which Assumptions did we use in the argument? Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

16 FWT cts. Which Assumptions did we use in the argument? completeness of markets Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

17 FWT cts. Which Assumptions did we use in the argument? completeness of markets A.1 (local non satiation) Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

18 FWT cts. Which Assumptions did we use in the argument? completeness of markets A.1 (local non satiation) free disposal Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

19 FWT cts. Which Assumptions did we use in the argument? completeness of markets A.1 (local non satiation) free disposal Is A.1 necessary? Counterexample: Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

20 FWT cts. Which Assumptions did we use in the argument? completeness of markets A.1 (local non satiation) free disposal Is A.1 necessary? Counterexample: Is completeness of markets necessary? Counterexamples: - economy where markets for some of the L commodities are not open. - economy with externalities: Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

21 FWT cts. Which Assumptions did we use in the argument? completeness of markets A.1 (local non satiation) free disposal Is A.1 necessary? Counterexample: Is completeness of markets necessary? Counterexamples: - economy where markets for some of the L commodities are not open. - economy with externalities: completeness - and hence e ciency - can be restored in this case by suitably expanding set of markets: Lindahl Equilibria. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

22 Gains from Trade Consider two countries: country A with H A consumers and country B with H B consumers (and the same L goods in each of them) Autarky equilibrium: p A, x Aaut such that country A markets (where only consumers of that country can trade) clear and p B, x Baut such that country B markets (where only country B consumers can trade) clear: H A x ha h A =1 H A ω ha ; h A =1 H B x hb h B =1 H B ω hb h B =1 Free trade equilibrium: p, x A, x B such that international markets (where consumers of both countries trade) clear: H A x ha + h A =1 H B x hb h B =1 H A h A =1 ω ha + H B h B =1 ω hb Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

23 Gains from Trade cts. Compare welfare under autarky and free trade (consider agents in one country): can everybody gain moving to free trade? Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

24 Gains from Trade cts. Compare welfare under autarky and free trade (consider agents in one country): can everybody gain moving to free trade? yes Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

25 Gains from Trade cts. Compare welfare under autarky and free trade (consider agents in one country): can everybody gain moving to free trade? yes can everybody lose moving to free trade? Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

26 Gains from Trade cts. Compare welfare under autarky and free trade (consider agents in one country): can everybody gain moving to free trade? yes can everybody lose moving to free trade? no Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

27 Gains from Trade cts. Proof. Compare welfare under autarky and free trade (consider agents in one country): can everybody gain moving to free trade? yes can everybody lose moving to free trade? no Suppose not: U ha (x haaut ) U ha (x ha ) for all h A = 1,.., H A (with a strict inequality for some h A ). Then p x haaut p x ha, with a strict inequality for some h A. Since p x ha = p ω ha for all h A (again assuming A.1), summing over h A yields p h A x haaut > p h A ω ha. But this contradicts feasibility of x A : h A x haaut h A ω ha. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

28 Gains from Trade cts. Still can say little on distribution of gains and losses. Can we nd a system of lump sum net transfers t ha, t hb to ha h B the agents in each country such that: (i) budget balance holds in each country ( h A t ha = 0, h A t hb = 0), and (ii) all agents (weakly) gain when going from autarky to free trade? Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

29 Gains from Trade cts. Still can say little on distribution of gains and losses. Can we nd a system of lump sum net transfers t ha, t hb to ha h B the agents in each country such that: (i) budget balance holds in each country ( h A t ha = 0, h A t hb = 0), and (ii) all agents (weakly) gain when going from autarky to free trade? yes (Grandmont - Mc Fadden 1972) Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

30 Gains from Trade cts. Proof. Still can say little on distribution of gains and losses. Can we nd a system of lump sum net transfers t ha, t hb to ha h B the agents in each country such that: (i) budget balance holds in each country ( h A t ha = 0, h A t hb = 0), and (ii) all agents (weakly) gain when going from autarky to free trade? yes (Grandmont - Mc Fadden 1972) [sketch] Set t ha = x haaut ω ha, t hb = x hbaut ω hb for all h A, h B. Budget balance then clearly holds (since x A, x B satisfy market clearing in each country). The Pareto improvement property then follows from the individual rationality of the competitive equilibrium (with free trade, relative to the after transfers endowment, given by x haaut for any h A and by x hbaut for any h B ). Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

31 SWT Second Welfare Theorem: Assume individual preference are strictly monotone and convex. Then every Pareto e cient allocation x can be decentralized as a competitive equilibrium with lump sum transfers. That is, for any Pareto e cient x, we can nd some lump sum transfers t h 2 R L, h = 1,..H, such that h t h = 0 and the economy where consumers have initial endowment ω h + t h has a competitive equilibrium given by p, x. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

32 SWT cts. Proof. Set t h = x h ω h for all h. Consider the following two sets: the singleton fωg, where ω h ω h = h x h and V h SP h (x h ) (the sum of the strictly preferred sets of each individual, evaluated at their consumption at the PE allocation). The second set is also convex (by the convexity of individual preferences). Hence, by the Separating Hyperplane Theorem 9p 2 R L, p 6= 0 and c 2 R: p z c 8z 2 V and p ω c. By strict monotonicity, ω 2 cl fv g, hence p ω = c. Also, for any ˆx h h x h (again by strict monotonicity and continuity of preferences) we have p ˆx h p x h for all h [why?]. Thus p, (x h ) h is a quasi-equilibrium (that is, at p, x h solves the expenditure minimization problem for all h and markets clear). Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

33 SWT end Proof. (cts.) If p x h > 0, then p ˆx h > p x h, that is p, (x h ) h is also an equilibrium.[why?] The strict monotonicity of preferences then implies that p 0 so that x h > 0 ) p x h > 0. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

34 Existence of Equilibria Competitive equilibrium prices are solutions of the following system of equations (under strict monotonicity): z(p; ω h ) x h (p, p ω h ) ω h = 0 h h h L equations in L unknowns (p). By Walras law (assuming A.1), p x h (p, p ω h ) h! ω h = 0 for all p h at most L 1 equations are independent (can always omit market clearing equation for one market). By homogeneity of degree zero in p of individual, and hence aggregate demand, prices can always be normalized, e.g. ( ) p 2 L 1 p 2 R L + : p l = 1 l : L-simplex (compact ad convex) Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

35 - Moreover, some boundary behavior (z 1 (p 1 ) < 0 for p 1 1 and z 1 (p 1 ) > 0 for p 1 0) also needed for existence Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48 Existence of Equilibria cts. Equilibrium equations can thus be always reduced to L 1 equations in L 1 unknowns. Since equations are typically nonlinear, having number of unknowns less or equal than number of independent equations does not ensure a solution exists To illustrate this, consider case L = 2: su ces to consider the function z 1 (p 1 ) - Illustrate graphically that if z 1 (p 1 ) is not continuous, existence may fail Note: what matters is continuity of aggregate demand. Can this be continuous when individual demand is not continuous? Yes, when we have economies with large number (in nitely many) consumers.

36 Existence of Equilibria cts. To show existence will use following Fixed Point Theorem (Brouwer): If the map f : S! S is continuous, and the set S is convex and compact, there exists a xed point x : f (x ) = x. Simple illustration of result when S = [0, 1]: As an immediate application of this result can show: Theorem Let z : L 1! R L be a continuous function, such that p z(p) = 0 for all p. Then there exists p such that z(p ) 0. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

37 Existence of Equilibria cts Proof. p Let ϕ l (p) = l + max f0, z l (p)g L, j = 1,..L j=1 [p j + max f0, z j (p)g] Note that L 1 is convex and compact, and ϕ : L 1! L 1. Hence by above FPT, there is a xed point p : p l = L j=1 pl + max f0, z l (p )g h i, j = 1,..L. Thus pj + max f0, z j (p )g z l (p )p l L p j + max f0, z j (p )g = z l (p )pl + z l (p ) max f0, z l (p )g j=1 Summing over l yields: 0 = z l (p ) max f0, z l (p )g ) z l (p ) 0 for all l. l Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

38 Existence of Equilibria cts p such that z(p ) 0 and Walras law (p z(p ) = 0) imply that z l (p ) < 0 requires pl = 0, but this is impossible under strict monotonicity. Hence: z(p ) = 0 To properly claim existence of a competitive equilibrium need to face one last problem: under strict monotonicity, consumers demand is not de ned for prices on the boundary of L 1 (when the price of some good is zero). ) use a limit argument (consider z : L ε 1! R L, for ε > 0: L ε 1 p 2 R L + : l p l = 1, p l ε for all l, and take limit as ε! 0) Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

39 Other Existence Arguments - With strictly convex preferences, a PE allocations maximizes, as we saw (over the set of feasible allocations), the linear social welfare function max x h ξ h U h (x h ) for some welfare weights (ξ h ) H h=1. - Under the assumptions of the SWT, any PE allocation satis es all the conditions for a competitive equilibrium except, possibly, the budget equation. Let p(ξ) be the price vector supporting the PE allocation (shadow price of resource constraints) identi ed by the vector of weights ξ. p(ξ) is also a competitive equilibrium price vector if: p(ξ)(x h (ξ) ω h ) = 0 for all h = 1,.., H This is a system of H 1 independent equations in H 1 unknowns (normalized weights ξ), convenient when L is large relative to H (e.g. in dynamic economies). Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

40 How many Equilibria can there be? Back to the case L = 2: simple graphical argument illustrates that: 1 there can be many competitive equilibria, also a continuum of them; Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

41 How many Equilibria can there be? Back to the case L = 2: simple graphical argument illustrates that: 1 there can be many competitive equilibria, also a continuum of them; 2 situations with a continuum of equilibria are non robust to small perturbations of the aggregate excess demand function; Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

42 How many Equilibria can there be? Back to the case L = 2: simple graphical argument illustrates that: 1 there can be many competitive equilibria, also a continuum of them; 2 situations with a continuum of equilibria are non robust to small perturbations of the aggregate excess demand function; 3 uniqueness holds if demand is downward sloping. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

43 Local Uniqueness and Comparative Statics Can indeed show (using di erential topology arguments, when utility functions are such that demand functions are continuous and di erentiable) that, for almost all economies (identi ed by endowment distribution (ω h ) H h=1 ), competitive equilibria are locally isolated and - the L 1 L 1 matrix jd ˆp ẑj 6= 0, where ˆp = (p 1,.., p L 1 ), ẑ = (z 1,.., z L 1 ), and p L = 1. Hence we can do comparative statics analysis: study how competitive equilibrium prices and allocations change in response to a change in the parameters, e.g. in the fundamentals of the economy, given by consumers preferences and endowments, or in policy parameters (taxes,..). Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

44 Comparative Statics Consider an in nitesimal change of the fundamentals, e.g. of the endowment of type 1 agents: dω 1. The e ect on equilibrium prices can be obtained by applying the IFT to the system of (L 1) equilibrium equations (in L 1 unknowns): h h ˆx h (p, p ω h ) ω hi = 0 if the matrix D ˆp ẑ = D ˆp ˆx h + D m ˆx h ˆω h T h is invertible. In that case we have h D ˆω 1 ˆp = (D ˆp ẑ) 1 d ˆω 1 D m ˆx 1 ˆp T d ˆω 1i Hence comparative statics e ects crucially depend on D ˆp ẑ, but we saw this matrix can be essentially arbitrary when H is su ciently large! Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

45 When is the equilibrium unique? when the initial endowment distribution is Pareto e cient: only equilibrium is no trade x h = ω h for all h Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

46 When is the equilibrium unique? when the initial endowment distribution is Pareto e cient: only equilibrium is no trade x h = ω h for all h when aggregation holds (a representative consumer exists) for all p and for given (m h ) H h=1 Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

47 When is the equilibrium unique? when the initial endowment distribution is Pareto e cient: only equilibrium is no trade x h = ω h for all h when aggregation holds (a representative consumer exists) for all p and for given (m h ) H h=1 when an appropriate generalization of the downward sloping demand condition to the case L > 2 (LOD) holds. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

48 Law of Demand and Uniqueness 1 If aggregate demand satis es WARP - weaker than LOD - (and equilibria are locally isolated) [z(p) = 0 and z(p 0 ) = 0 imply, by WARP, that z(αp + (1 α)p 0 ) = 0 for all α. This follows from the fact that Walras law implies that either p z(αp + (1 α)p 0 ) 0 or p 0 z(αp + (1 α)p 0 ) 0. But also (αp + (1 α)p 0 ) z(p) 0, (αp + (1 α)p 0 ) z(p 0 ) 0, hence by WARP z(αp + (1 α)p 0 ) = z(p) for all α. But this contradict the fact that equilibria are locally isolated.] Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

49 Law of Demand and Uniqueness cts 2. If aggregate demand satis es the gross substitute (GS) property: for any p, p 0 such that p 0 l > p l and p 0 j = p j for j 6= l, we have: z j (p 0 ) > z j (p) for all j 6= l (that is, an increase in the price of one good increases the demand in all other goods), then there is at most one competitive equilibrium. [Suppose z(p) = z(p 0 ) = 0. By homogeneity z(αp) = 0 for all α > 0. Set ᾱ = max l p 0 l /p l, so that ᾱp p 0, applying GS yields a contradiction] Note that GS implies that LOD holds at any equilibrium price [GS holds for instance for Cobb Douglas and CES utility functions with σ < 1] Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

50 Strategic Foundations Consider market games: cooperative: core,... Here will consider a noncooperative market game (Shapley-Shubik): each consumer h chooses supply sl h 0 and bid bl h 0 for each commodity l = 1,.., L 1, such that: sl h ω h l for all l < L, L l=1 1 bh l ω h L given the bids and supplies of all consumers, prices (in terms of commodity L) are determined as follows: and hence consumption levels: p l (b, s) = h b h l h s h l xl h = ω h l sl h + bh l, for l = 1,.., L 1 p l (b, s) x h l = ω h l L 1 b h L 1 l + sl h p l (b, s) l=1 l=1 Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

51 Market games - Equilibrium: Nash equilibrium of the market game described need to specify level of prices in closed markets, where h bl h h sl h are zero. and/or prices move against trade of an agent: the more he bids for a good (say l), the lower will be 1/p l (b, s) Budget set for agent h is convex (when all markets are not closed ). as H!, if both b h l / h b h l and s h l / h s h l! 0 the budget set becomes linear in the limit and the Nash equilibrium coincides with the competitive equilibrium. Price taking behavior justi ed when the market share of each agent is negligible Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

52 Economy under Uncertainty individual endowment: eω h, random variable, with ( nite) support ω h (1),.., ω h (S) 2 R S +, for all h [consider, for simplicity, case where L = 1] State space S = f1,.., Sg, π = (π(1),.., π(s)) allocation: ex h, also a random variable, with support x h = x h (1),.., x h (S) 2 R S + (de ned on the same state space), for all h Preferences, now de ned over random consumption bundles: U h : ex! R e.g. VNM preferences (expected utility): s π(s)u h (x(s)). Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

53 Allocation of Risk Feasible allocation: an allocation x = (.., ex h,..) such that: h x h (s) ω h (s) for all s = 1,.., S h Uncertainty only a ects the fundamentals of the economy via the agents endowments (no preference shocks) Economic Problem: allocation of (income) risk among the consumers Preferences describe individual attitude towards risk: with VNM, strict convexity of preferences (u h : R +! R strictly concave) implies agent is risk averse, i.e. Eex h ex Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

54 E cient allocations An allocation x = (.., ex h,..) is Pareto e cient if: if it is feasible there is no other feasible allocation that Pareto dominates it Examples (assuming VNM preferences, S = 2, H = 2): Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

55 E cient allocations An allocation x = (.., ex h,..) is Pareto e cient if: if it is feasible there is no other feasible allocation that Pareto dominates it Examples (assuming VNM preferences, S = 2, H = 2): no aggregate risk: h ω h (s) = ω for all s: full risk sharing is possible. Is it also optimal? Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

56 E cient allocations An allocation x = (.., ex h,..) is Pareto e cient if: if it is feasible there is no other feasible allocation that Pareto dominates it Examples (assuming VNM preferences, S = 2, H = 2): no aggregate risk: h ω h (s) = ω for all s: full risk sharing is possible. Is it also optimal? yes, if agents have identical beliefs: π h = π for all h Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

57 E cient allocations An allocation x = (.., ex h,..) is Pareto e cient if: if it is feasible there is no other feasible allocation that Pareto dominates it Examples (assuming VNM preferences, S = 2, H = 2): no aggregate risk: h ω h (s) = ω for all s: full risk sharing is possible. Is it also optimal? yes, if agents have identical beliefs: π h = π for all h no, if agents have di erent beliefs π 1 6= π 2 (!some betting/speculative trade is e cient in this case) Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

58 E cient allocations An allocation x = (.., ex h,..) is Pareto e cient if: if it is feasible there is no other feasible allocation that Pareto dominates it Examples (assuming VNM preferences, S = 2, H = 2): no aggregate risk: h ω h (s) = ω for all s: full risk sharing is possible. Is it also optimal? yes, if agents have identical beliefs: π h = π for all h no, if agents have di erent beliefs π 1 6= π 2 (!some betting/speculative trade is e cient in this case) aggregate risk: h ω h (1) 6= h ω h (2): full risk sharing not feasible, must also allocate aggregate risk. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

59 Markets Individual preferences (under VNM): s π(s)u h (x(s)) Complete markets: there exists one market for each s = 1,..S where agents can trade claims for delivery of the commodity contingent on the realization of that state + p = (p(1),.., p(s)) Timing of markets also important: markets open before realization of the uncertainty Consumers choice problem: max x s.t. p x p ω π(s)u h (x(s)) s formally analogous to the one in deterministic case Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

60 Competitive equilibria A competitive equilibrium is given by a price vector p and an allocation x if the allocation solves the choice problem of every consumer h at the prices p and the allocation is feasible. Also formally the same as in deterministic case. Thus same properties hold: existence, FWT, SWT,... Examples again: with no aggregate risk and identical beliefs equilibrium is given by: p(s) = π(s) for all s, x h (s) = x h = s π(s)ω h (s) for all h with no aggregate risk and π 1 (1) > π 2 (1): π 1 (1) > p(1) > π 2 (1) and x 1 (1) > x 1 (2) with aggregate risk h ω h (1) > h ω h (2), same beliefs: p(1) < π(1) Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

61 Introducing Time/Sequential Trades Asset markets open before the realization of the uncertainty, for the trades of claims for the delivery of income (assets) contingent on the realization of each individual state. E.g.: q s : price of a claim promising the future delivery of one unit of income if and only if state s is realized (contingent, or Arrow security), s 2 S The subsequent period agents receive the net payo on their portfolio and use their income to buy the good to be consumed (or trade again if L > 1 and/or T > 1 (dynamic economy)) Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

62 Asset Markets and Sequential Trades Budget constraint becomes: - for asset markets, open before realization of uncertainty: q s θ s = 0 s where θ s denotes, if > 0 (resp. < 0), a long (resp. short) position in contingent security s - after realization of uncertainty: (2a) x(s) ω h (s) θ s for each state s = 1,..S (3) Consumer s choice problem: choose θ 2 R S, x 2 R S + so as to maximize h s utility (say s π(s)u h (x(s))) subject to the two above constraints Note: Utility for asset holdings is only indirect Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

63 Equilibria with sequential trades Market clearing (feasible allocation): for assets: for commodities: x h (s) h θ h s = 0 for each s h ω h (s) 0 for each s Competitive equilibrium with sequential trades: an array of asset prices q, a consumption allocation x and an asset allocation θ =.., θ h,.. if for all h, x h 2 R S +, θh 2 R S solve above consumer s choice problem, given q. both the allocation of assets θ and of consumption goods in every state x are feasible Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

64 Arrow equivalence result Proof. Theorem (Arrow 1953): The set of competitive equilibrium allocations obtained with complete markets for contingent commodities (AD equilibria) coincides with the set of competitive equilibrium allocations obtained with sequential trades and S Arrow securities (sketch) Let p, x be an AD equilibrium. Set q s = p s, for all s. It is straightforward to verify that, with these prices, the set of budget feasible consumption bundles with the two market structures coincide, for all consumers. Hence their consumption choice will also be the same. Viceversa, let q and x, θ be a competitive equilibrium with sequential trades and complete Arrow securities. Set p s = q s for all s. Again it can be easily veri es that, with these prices, the set of budget feasible consumption bundles with the two market structures are the same. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

65 General asset structures Instead of the special case of Arrow securities, suppose there are J arbitrary assets. Each asset j is identi ed by its (unit) return, a random variable er j, also described by its support r j = (.., r j (s),..) T (e.g.: bonds, equity, derivatives,...) Consumer s budget constraints become: q s θ s = 0 (unchanged) s x(s) ω h (s) r j (s)θ j for each state s = 1,..S j Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

66 Complete vs. Incomplete Markets Let R be the S J matrix of the payo s of the existing assets, with generic element r j (s) (in the previous case, with Arrow securities, R = I ). When R has full rank S, we say asset markets are complete (for any return pro le y 2 R S, there is a portfolio θ such that Rθ = y) Equivalence result of previous page extends to any asset structure with complete markets. Hence equilibrium allocations are Pareto e cient,... When rank of R is less than S, we say asset markets are incomplete: In that case, sequence of budget constraints can no longer be reduced to a single intertemporal budget constraint. Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

67 Incomplete Markets With incomplete markets need to study the solutions of the agents portfolio and consumption choice problem. Its FOCs (for an interior solution) are: They imply that: π(s)du h (x(s)) = λ 1 (s) λ 0 q = r(s)λ 1 (s) = R T λ 1 s plus the budget constraints the MRS between the consumption between any pair of states s, s 0, u h (x (s))/ x u h (x (s 0 ))/ x, is typically di erent across consumers )allocation of risk across consumers is (typically) not e cient Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48

68 Incomplete Markets Example: let L = 1, S = 3, H = 2 consumers with identical beliefs, no aggregate risk, ω 1 (s) 6= ω 1 (s 0 ) for all s 6= s 0. Suppose: R = Then at any allocation attainable with these assets we have thus e ciency cannot be attained. u 10 (x 1 (3)) u 10 (x 1 (2)) 6= u20 (x 2 (3)) u 20 (x 2 (2)) Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, / 48