General Equilibrium under Uncertainty

General Equilibrium under Uncertainty

The Arrow-Debreu Model General Idea: this model is formally identical to the GE model commodities are interpreted as contingent commodities (commodities are contingent to a state of nature, date, location) Definition of the list of commodities S states of nature s = 1,..., S L physical commodities l = 1,..., L LS contingent commodities (index ls) The Arrow Debreu model = the GE model with LS goods are numbered ls) goods (the

Interpreting the model: what does trading contingent goods mean? 2 periods: at t = 0, agents ignore s, they trade contingent goods to exchange x ls units of good ls against x l s units of good l s = to sign a contract committing to deliver x ls units of physical good l if state s occurs at t = 1 in exchange of receiving x l s units of physical goods l if state s occurs at t = 1 At t = 1, no contract is signed (markets are closed): the state of nature s becomes public, contracts contingent to state s are executed (= physical goods are delivered and consumed), contracts contingent to other states are destroyed (they have no value)

endowment (ω ils ) ls of agent i: at the beginning of date 1, in state s, agent i receives a quantity ω ils of physical good l optimal demand (x ils ) ls of agent i is a decision taken at t = 0: at t = 0, agent decides that, at t = 1, if state s occurs, he will consume x ils units of physical good l (at t = 0, he signs a contract committing him to buy z ils = x ils ω ils units of physical good l if state s occurs at t = 1)

Trading contingent goods allows agents to transfer wealth across states The demand of i maximizes his utility function subject to the budget constraint p.x = p.ω, that is: p ls x ils = p ls ω ils l,s l,s This rewrites t s = 0 s where t s is the wealth that needs to be transferred to state s: t s = p ls x ils p ls ω ils (t s 0 or t s 0) l }{{} value of the consumption in s l }{{} value of the endowment in s

one can have a model with more than 2 periods and 1 state of nature or more as well (all the contingent goods are traded at t = 0 and not later, these are commitments to deliver physical goods at a certain date t in a certain state s in exchange of receiving something at a date t 1 in a state s ) Interest of the model: to discuss risk sharing and intertemporal trades (saving, borrowing) with all the tools (concepts and theorems) of the GE Theory (for example, equality between the MRS at equilibrium across all dates and states)

Remark on the utility of i: This is a function with LS variables x ils One can assume an expected utility form: at t = 0, agent i assigns probability π is to state s v i is a function with L variables ( = a bundle of L physical goods) the agent chooses his demand by maximizing u i (x i11,..., x ils ) = s π is v i (x i1s,..., x ils ) v i can depend on s as well: max s π is v i (x i1s,..., x ils, s)

About production of contingent goods Example with L = S = 2 (2 physical goods, 2 states of nature), y = (y 1a, y 1b, y 2a, y 2b ) the firm transforms l = 1 into l = 2, whatever s = a, b is (production function f ). The production set is { y IR 4 /y 1a = y 1b, y 2a = y 2b, y 2a f ( y 1a ) } the firm s technology is not the same in the 2 states (2 production functions f a and f b ) and the firm decides the amount of input before s is revealed. The production set is { y IR 4 /y 1a = y 1b, y 2a f a ( y 1a ), y 2b f b ( y 1a ) }

Arrow Debreu Equilibrium This is exactly the usual equilibrium: an allocation (x 1,..., x I ) IRILS +, (y 1,..., y J ) IRJLS, prices p IR LS + such that Individual optimality i, x i solves max u i (x i ) subject to the budget constraint p.x i p.ω i + j θ ij π j j, y j solves max p.y j subject to y j Y j (denote π j = p.y j ) Market clearing: l, s, i x ils = i ω ils + j y jls

The Welfare Theorems apply (1st Theorem means that risk sharing across states is efficient) The existence theorem applies as well (convexity of preferences and production sets implies existence of an equilibrium)

Risk sharing: an example an exchange economy with 2 consumers i = A, B, L = 1, S = 2 (2 contingent goods: wealth in state s = 1, 2) utility has an EU form: for i = A, B, denote with u i C 2 (u < 0 < u ) 3 cases π 1i u i (x 1i ) + π 2i u i (x 2i ) no aggregate uncertainty (ϖ 1 = ϖ 2 ) and objective probabilities (π 1A = π 1B ) no aggregate uncertainty and subjective probabilities (π 1A < π 1B ) aggregate uncertainty (ϖ 1 < ϖ 2 ) and objective probabilities

Consider an interior equilibrium: MRS A = MRS B. This implies: Case ϖ 1 = ϖ 2 and π 1A = π 1B complete insurance (xi1 = x i2 ) p2 p 1 = π2 π 1 Case ϖ 1 = ϖ 2 and π 1A < π 1B partial insurance xa1 < x A2 and x B1 > x B2 π 2B π 1B < p2 p 1 < π 2A π 1A Case ϖ 1 < ϖ 2 and π 1A = π 1B xa1 < x A2 and x B1 < x B2 p2 π 2 < p1 π 1 The example generalizes to an arbitrary number of states and consumers

Sequential Trade Preliminary remark: Mas-Colell et alii present an example of sequential trade (19D) before the introduction of asset markets (19E). The case carried in 19D (Definition 19D1 and Proposition 19D1) is an example of a model with asset markets (where the Arrow assets only are available) This is why I skip most of this section. I give 2 remarks only

Remark 1 Opening of the markets at t = 1 is useless the above interpretation of the AD model: contingent goods markets at t = 0 and no market at t = 1 one could open the markets at t = 1: spot markets for the L physical goods 1st Welfare Theorem implies that, at equilibrium, no additional trade occurs at t = 1

An alternative interpretation is: there would be too many markets that is: agents could trade good ls against good l s at t = 1 (if state s occurs) instead of trading at t = 0 when the spot markets exist, not all the markets at t = 0 are necessary for the equ alloc to be PO. Markets at t = 0 are required to transfer wealth across states only (not to exchange goods within the same state ). This is what the example in 19D shows: trading the S contingent goods 1s at t = 0 is enough ( one good/state ). In the context of asset markets, these goods are named Arrow assets

Remark 2: rational expectations when trade is sequential, there are opened markets at differents dates Apparently, the decision that is made at t = 0 concerns goods traded at t = 0 only But, to make an optimal choice at t = 0, the agent needs to determine at the same moment what he will do at t = 1 (the decisions made at t = 0 and t = 1 both concern max u i ) Hence, at t = 0, he needs to know the prices of the goods traded at t = 1 The AD model assumes that the agents perfectly anticipates the prices. Perfect expectations are the so-called rational expectations in this model (in other models, rational expectations are not perfect: some uncertainty remains) Rational Expectations are assumed in most definitions of equilibrium. A general definition is beyond the scope of this course

Asset Markets Consider an exchange economy with I consumers, L physical goods, S states of nature, K assets at t = 0, s is unknown and assets are exchanged at t = 1, s becomes public, asset payoffs are paid, goods are exchanged What is an asset? a title (price q k ) to receive an amount r sk 0 units of good 1 at t = 1 in state s this is a real asset (payoff in good 1 only is for convenience) nominal assets r sk < 0 could be considered as well

Examples of assets safe asset r = (c,..., c) the S Arrow assets r s = (0,..., 0, 1, 0,..., 0) (r ss = 1, r sk = 0 otherwise) option (example of a derivative asset) with strike price c, primary asset r k call : r option = (max (0, r ks c)) 1 s S put : r option = (max (0, c r ks )) 1 s S

The agent chooses at t = 0 a portfolio of assets z i = (z i1,..., z ik ) IR K a commodity bundle x is = (x is1,..., x isl ) IR L + in each state s s.t. s, l max s q k z ik = 0 k p ls x ils = l π is u i (x is1,..., x isl ) p ls ω ils + p 1s r sk z ik Typically, z k 0 for some k and z k 0 for some other k no endowment of asset (this assumption for convenience only) usual endowment ω ils of physical good l in state s (received at the beginning of t = 1) k

Radner Equilibrium An allocation (z, x) IR KI IR LSI + and prices (q, p) IR K+LS + such that: Individual optimality: i, (z i, x i ) solves the max problem on the previous slide Market clearing: k, i z ik = 0 l, s, i x ils = i ω ils

Comments: p s : spot prices in state s w.l.o.g. s, p s1 = 1 equilibrium of plans, prices and price expectations (portfolio and consumption plans, the definition includes an assumption about the price expectations : see the previous slide on rational expectations )

Fundamental property 1 of the equilibrium asset prices At equilibrium, q is arbitrage-free The converse implication is not true: an arbitrage-free price vector q is not always an equilibrium price vector

Proof An arbitrage portfolio is z i such that q.z i 0, s, k r sk z ki 0 with strict inequality for (at least) one s If there is an arbitrage portfolio, then no agent i has an optimal decision: consider a decision (x i, z i ) z i + zi gives a larger wealth in every state s there is x i such that u i (x is1,..., x isl ) > u i (x is1,..., x isl ) in every s (given that u i is increasing) loosely speaking, i wants to buy the portfolio λzi with λ = + in order to get an infinite wealth in every state s (so that no market clearing is possible - assets or goods -)

Fundamental property 2 of the equilibrium asset prices At equilibrium, there are (µ 1,..., µ S ) 0 such that, for every asset with returns r k q k = µ s r sk µ s is the price of the Arrow asset s s Proof: Farkas lemma (derived from a separating hyperplane theorem, see Mas-Colell et alii)

Proof consider the space IR K { z IR K /q.z < 0 } is a convex set the intersection of the S convex sets (one for each asset) { z IR K / s r skz k 0 } is a convex set the 2 sets do not intersect (the asset prices are arbitrage-free) there is (µ 1,..., µ S ) 0 such that q 1. q K = s µ s r s1. r sk

The fundamental characteristic of the market structure It is either complete or incomplete Definition: an asset structure (K assets, associated with a S K return matrix R = (r sk ) s,k ) is complete iff the rank of R is S. Comments: S is the maximal possible rank for a S K matrix this means that there are S linearly independent assets (example: the S Arrow assets) S assets are enough to get a complete market structure (but the markets may be incomplete even if K > S) With S linearly independent assets, further assets are redundant: their r k is a linear combination of the r k of the S first assets

Interpretation: all the transfers of wealth across states are feasible Budget constraint over the S states (this is the BC of the associated Arrow Debreu economy) p ls x ils = p ls ω ils l,s l,s define a transfer of wealth to state s s, t s = l p ls x ils l p ls ω ils question is: are these transfers feasible through an appropriate portfolio z i, that is: is there z i such that s, p 1s r sk z ik = t s k if the return matrix R has rank S, then the answer is yes the remaining question is: does z i satisfy q.z i 0? (see next slides)

A formal statement of the above interpretation This statement = 2 converse implications (stating that Radner equilibrium = Arrow Debreu equilibrium) an Arrow Debreu economy with no asset and LS contingent goods can be associated with every economy with complete asset markets and sequential trades, conversely, with an Arrow Debreu economy, we can associate a market structure (typically the S Arrow assets) and consider the economy with sequential trade where assets are traded at t = 0 and the L physical goods are traded at t = 1 (once the state s is revealed)

Implication 1: Consider an economy with complete asset markets. If (x, z, p, q) IR + LSI IR KI IR + LS IR + K is a Radner equilibrium (normalization condition: s, p 1s = 1), then there is (µ 1,..., µ S ) IR + S such that (x, µ 1 p 1,..., µ S p S ) IR + LSI IR + LS is an Arrow Debreu equilibrium (p s IR + S is the vector of prices of contingent goods ls, µ s is the value of state s, that is the price of the Arrow asset s) Fundamental consequence of Implication 1: the 2 welfare theorems apply when asset markets are complete, the allocation x of a Radner equilibrium is PO a PO allocation x is the allocation of a Radner equilibrium (with appropriate wealth transfers)

Implication 2: If (x, p) IR LSI + IR LS + is an Arrow Debreu equilibrium, then there is a market structure with K assets (of returns r k ) and there are portfolios z IR KI and asset prices q IR K + such that (x, z, p, q) is a Radner equilibrium Fundamental consequence of Implication 2: the existence theorem applies when the asset markets are complete, there is a Radner equilibrium (given concavity of the u i )

Proofs Consider the 2 budget sets (p R 1s = 1) B R = B AD = { x i IR + LS / z i IR K /q.z i 0 and s, l pr ls x ils l pr ls ω ils + k r skz ik x i IR LS + / l,s pls AD x ils l,s p AD ls ω ils } We show that, for properly chosen p AD and p R, these 2 sets coincide

Inclusion B R B AD consider the µ s such that q k = s µ sr sk for x i B R ( ) µ s pls R x ils ( µ s pls R ω ils + ) r sk z ik s l s l k ( ) µ s pls R x ils ( ) µ s pls R ω ils + ( ) µ s r sk z ik l,s l,s k s hence, with pls AD = µ s pls R, pls AD x ils l,s l,s that is: x i B AD pls AD ω ils

Inclusion B AD B R consider the S Arrow assets (complete asset markets) define q s = p1s AD for x i B AD, define check that q.z i 0 and check that and p R ls = pad ls ( s, z is = 1 p1s AD pls AD x ils l l /p AD 1s (notice p R 1s = 1) ) pls AD ω ils s, l p R ls x ils = l p R ls ω ils + k r sk z ik that is: x i B R

About redundant assets If x is the allocation of a Radner equilibrium with an asset structure associated with a return matrix R, then x is the allocation of a Radner equilibrium with any other asset structure associated with a return matrix R such that ranger = ranger range of a (return) S K matrix: ranger = { v IR S /v = Rz, z IR K } redundant assets can be deleted without changing the allocation of Radner equilibrium

About incomplete markets When markets are incomplete there can be no equilibrium equilibrium can be suboptimal equilibrium is sometimes not even constrained optimal (constrained optimality: Pareto optimality among the allocations x that are feasible given the asset structure)

The end of the chapter