General Equilibrium under Uncertainty

The Arrow-Debreu Model

General Idea:

I this model is formally identical to the GE modelI commodities are interpreted as contingent commodities

(commodities are contingent to a state of nature, date,location)

Definition of the list of commodities

I S states of nature s = 1, ...,SI L physical commodities l = 1, ..., LI LS contingent commodities (index ls)

The Arrow Debreu model = the GE model with LS goods (thegoods are numbered ls)

Interpreting the model: what does trading contingent goodsmean?

2 periods:

I at t = 0, agents ignore s, they trade contingent goodsI to exchange xls units of good ls against xl s units of good l s

= to sign a contract committing to deliver xls units of physicalgood l if state s occurs at t = 1 in exchange of receiving xl s

units of physical goods l if state s occurs at t = 1I At t = 1, no contract is signed (markets are closed): the state

of nature s becomes public, contracts contingent to state sare executed (= physical goods are delivered and consumed),contracts contingent to other states are destroyed (they haveno value)

I endowment (ils)ls of agent i : at the beginning of date 1, instate s, agent i receives a quantity ils of physical good l

I optimal demand (xils)ls of agent i is a decision taken at t = 0:at t = 0, agent decides that, at t = 1, if state s occurs, hewill consume xils units of physical good l (at t = 0, he signs acontract committing him to buy zils = xils ils units ofphysical good l if state s occurs at t = 1)

Trading contingent goods allows agents to transfer wealth acrossstates

I The demand of i maximizes his utility function subject to thebudget constraint p.x = p., that is:

l ,s

plsxils =l ,s

plsils

I This rewrites s

ts = 0

where ts is the wealth that needs to be transferred to state s:

ts =l

plsxils value of the consumption in s

l

plsils value of the endowment in s

(ts 0 or ts 0)

I one can have a model with more than 2 periods and 1 state ofnature or more as well (all the contingent goods are traded att = 0 and not later, these are commitments to deliver physicalgoods at a certain date t in a certain state s in exchange ofreceiving something at a date t 1 in a state s )

I Interest of the model: to discuss risk sharing and intertemporaltrades (saving, borrowing) with all the tools (concepts andtheorems) of the GE Theory (for example, equality betweenthe MRS at equilibrium across all dates and states)

Remark on the utility of i :

This is a function with LS variables xilsOne can assume an expected utility form:

I at t = 0, agent i assigns probability piis to state sI vi is a function with L variables ( = a bundle of L physical

goods)

I the agent chooses his demand by maximizing

ui (xi11, ..., xiLS) =s

piisvi (xi1s , ..., xiLs)

I vi can depend on s as well:

maxs

piisvi (xi1s , ..., xiLs , s)

About production of contingent goods

Example with L = S = 2 (2 physical goods, 2 states of nature),y = (y1a, y1b, y2a, y2b)

I the firm transforms l = 1 into l = 2, whatever s = a, b is(production function f ). The production set is{

y IR4/y1a = y1b, y2a = y2b, y2a f (y1a)}

I the firms technology is not the same in the 2 states (2production functions fa and fb) and the firm decides theamount of input before s is revealed. The production set is{

y IR4/y1a = y1b, y2a fa (y1a) , y2b fb (y1a)}

Arrow Debreu Equilibrium

This is exactly the usual equilibrium:

I an allocation (x1 , ..., xI ) IR ILS+ , (y1 , ..., yJ ) IRJLS ,

I prices p IRLS+such that

I Individual optimalityI i , xi solves max ui (xi ) subject to the budget constraint

p.xi p.i +j

ijpij

I j , yj solves max p.yj subject to yj Yj (denote pij = p.yj )I Market clearing:

l , s,i

xils =i

ils +j

yjls

I The Welfare Theorems apply (1st Theorem means that risksharing across states is efficient)

I The existence theorem applies as well (convexity of preferencesand production sets implies existence of an equilibrium)

Risk sharing: an example

an exchange economy with 2 consumers i = A,B, L = 1, S = 2 (2contingent goods: wealth in state s = 1, 2)utility has an EU form: for i = A,B, denote

pi1iui (x1i ) + pi2iui (x2i )

with ui C2 (u < 0 < u)

3 cases

I no aggregate uncertainty ($1 = $2) and objectiveprobabilities (pi1A = pi1B)

I no aggregate uncertainty and subjective probabilities(pi1A < pi1B)

I aggregate uncertainty ($1 < $2) and objective probabilities

Consider an interior equilibrium: MRSA = MRSB . This implies:

I Case $1 = $2 and pi1A = pi1BI complete insurance (xi1 = xi2)I p2

p1= pi2pi1

I Case $1 = $2 and pi1A < pi1BI partial insurance xA1 < xA2 and xB1 > xB2I pi2B

pi1B< p2p1 ui (xis1, ..., xisL) in every

s (given that ui is increasing)

I loosely speaking, i wants to buy the portfolio zi with = + in order to get an infinite wealth in every state s (sothat no market clearing is possible - assets or goods -)

Fundamental property 2 of the equilibrium asset prices

At equilibrium, there are (1, ..., S) 0 such that, for every assetwith returns rk

qk =s

srsk

I s is the price of the Arrow asset sI Proof: Farkas lemma (derived from a separating hyperplane

theorem, see Mas-Colell et alii)

Proof

I consider the space IRK

I{z IRK/q.z < 0} is a convex set

I the intersection of the S convex sets (one for each asset){z IRK/s rskzk 0} is a convex set

I the 2 sets do not intersect (the asset prices are arbitrage-free)I there is (1, ..., S) 0 such that q1...

qK

= s

s

rs1...rsK

The fundamental characteristic of the market structure

It is either complete or incomplete

Definition: an asset structure (K assets, associated with a S Kreturn matrix R = (rsk)s,k) is complete iff the rank of R is S .

Comments:

I S is the maximal possible rank for a S K matrixI this means that there are S linearly independent assets

(example: the S Arrow assets)

I S assets are enough to get a complete market structure (butthe markets may be incomplete even if K > S)

I With S linearly independent assets, further assets areredundant: their rk is a linear combination of the rk of the Sfirst assets

Interpretation: all the transfers of wealth across states are feasible

I Budget constraint over the S states (this is the BC of theassociated Arrow Debreu economy)

l ,s

plsxils =l ,s

plsils

I define a transfer of wealth to state s

s, ts =l

plsxils l

plsils

I question is: are these transfers feasible through an appropriateportfolio zi , that is: is there zi such that

s, p1sk

rskzik = ts

I if the return matrix R has rank S , then the answer is yesI the remaining question is: does zi satisfy q.zi 0? (see next

slides)

A formal statement of the above interpretation

I This statement = 2 converse implications (stating thatRadner equilibrium = Arrow Debreu equilibrium)

I an Arrow Debreu economy with no asset and LS contingentgoods can be associated with every economy with completeasset markets and sequential trades,

I conversely, with an Arrow Debreu economy, we can associatea market structure (typically the S Arrow assets) and considerthe economy with sequential trade where assets are traded att = 0 and the L physical goods are traded at t = 1 (once thestate s is revealed)

Implication 1: Consider an economy with complete asset markets.If (x, z, p, q) IRLSI+ IRKI IRLS+ IRK+ is a Radner equilibrium(normalization condition: s, p1s = 1), then there is(1, ..., S) IRS+ such that (x, 1p1, ..., SpS) IRLSI+ IRLS+ isan Arrow Debreu equilibrium (ps IRS+ is the vector of prices ofcontingent goods ls, s is the value of state s, that is the priceof the Arrow asset s)

Fundamental consequence of Implication 1: the 2 welfare theoremsapply

I when asset markets are complete, the allocation x of aRadner equilibrium is PO

I a PO allocation x is the allocation of a Radner equilibrium(with appropriate wealth transfers)

Implication 2: If (x, p) IRLSI+ IRLS+ is an Arrow Debreuequilibrium, then there is a market structure with K assets (ofreturns rk) and there are portfolios z

IRKI and asset pricesq IRK+ such that (x, z, p, q) is a Radner equilibrium

Fundamental consequence of Implication 2: the existence theoremapplies

I when the asset markets are complete, there is a Radnerequilibrium (given concavity of the ui )

Proofs

Consider the 2 budget sets (pR1s = 1)

BR =

{xi IRLS+ /zi IRK/q.zi 0

and s,l pRls xils l pRlsils +k rskzik}

BAD =

xi IRLS+ /l ,s

pADls xils l ,s

pADls ils

We show that, for properly chosen pAD and pR , these 2 setscoincide

Inclusion BR BADI consider the s such that qk =

s srsk

I for xi BRs

s

(l

pRls xils

)

s

s

(l

pRlsils +k

rskzik

)l ,s

(sp

Rls

)xils

l ,s

(sp

Rls

)ils +

k

(s

srsk

)zik

I hence, with pADls = spRls ,

l ,s

pADls xils l ,s

pADls ils

I that is: xi BAD

Inclusion BAD BRI consider the S Arrow assets (complete asset markets)I define qs = pAD1s and p

Rls = p

ADls /p

AD1s (notice p

R1s = 1)

I for xi BAD , define

s, zis = 1pAD1s

(l

pADls xils l

pADls ils

)

I check that q.zi 0I and check that

s,l

pRls xils =l

pRlsils +k

rskzik

I that is: xi BR

About redundant assets

If x is the allocation of a Radner equilibrium with an assetstructure associated with a return matrix R, then x is theallocation of a Radner equilibrium with any other asset structureassociated with a return matrix R such that rangeR = rangeR

I range of a (return) S K matrix:rangeR =

{v IRS/v = Rz , z IRK}

I redundant assets can be deleted without changing theallocation of Radner equilibrium

About incomplete markets

When markets are incomplete

I there can be no equilibriumI equilibrium can be suboptimalI equilibrium is sometimes not even constrained optimal

(constrained optimality: Pareto optimality among theallocations x that are feasible given the asset structure)

The end of the chapter