Jean-Marc Bottazzi
email: jean-marc@bottazzi.org

Jaime Luque
email: japluque@eco.uc3m.es

Mário R Pascoa
email: pascoa@fe.unl.pt

Mário R Páscoa

Documents de Travail du

Keywords: repo, short sale, bubble, repo specialness, Ponzi scheme, leverage. JEL classification numbers : C62, D52, D53, D90, G12

Securities markets theory includes repo and distinguishes shorting from issuing. Here we revisit whether trading alone can give rise to Ponzi schemes and rational bubbles. We show that under the same institutional arrangements that limit re-hypothecation (e.g., through segregated haircut rules or explicit leverage constraints on haircut collecting dealers), (1) trading Ponzi schemes are prevented without having to assume uniform impatience, (2) for securities in positive net supply, bubbles are ruled out under complete markets but may occur when markets are incomplete. We give an example of such a bubble, under a finite present value of wealth.

Introduction

This paper revisits speci…cally for the securities market two major issues in in-…nite horizon equilibrium theory: the occurrence of Ponzi schemes and of price bubbles. We do this by capturing the basic fact that in order to short sell a security, one has to borrow it. In this context, shorting and issuing are perfectly distinguished. This allows us to focus on the pure trading aspects of securities.

Rather than assuming solvency constraints or debt limits, we see how the way securities are traded already constrains the asymptotic values of security positions. Then, we address the problematic introduced by Santos and Woodford (1997), "the conditions under which asset prices in an inter-temporal competitive equilibrium are equal to" the expected discounted value of futures cash ‡ows.

It is our view that repo market theory is a core part of security markets theory, as borrowing and lending of securities cannot be ignored. Models have to keep track of both security positions and (title) possession out of a …nite pool.

Formally, a repo trade consists in a security sale under the agreement of a future repurchase at a predetermined date and price. It is a collateralized loan, where the security that serves as a collateral is kept by the creditor (the borrower of the security), who tends to use it, lending on or shorting, before the delivery is due.

A bit out of necessity (so bubbles are not merely killed by the …nite horizon!), we extend the literature on …nancial equilibrium that has incorporated the repo market in a general equilibrium model with …nite horizon (see Du¢ e (1996) and Bottazzi, Luque and Páscoa (2011a)). Thus, we add a market for funding securities in an in…nite horizon framework, absent in previous papers in the …eld (see Araujo, Páscoa and Torres-Martínez (2002), Hernández and Santos (1996), Magill and Quinzii (1994, 1996) and Santos and Woodford (1997)). Also, we have seen how Bottazzi, Luque and Páscoa (2011a) introduces possession (or specialness) value as part of the normal value of a security. Thus, we include possession value into the fundamental value of a security. And we will say that there is a bubble in a given security if its value exceeds the sum of the expected discounted value of its cash ‡ows plus its possession value.

When repo markets were ignored, the equilibrium literature had assumed two sorts of portfolio constraints, with quite di¤erent purposes. One, (a) a plain bounded short sales constraint, for equilibrium in …nite horizon economies, in order to overcome Hart's (1975) problem. Another, (b) constraints on the value of debt (bounding it uniformly or relating it to future wealth) that managed to avoid Ponzi schemes, when uniform impatience was assumed. An exception was the model by Araujo, Páscoa and Torres-Martínez (2002) with collateralized promises, such as mortgages, where the same form of secured borrowing served both purposes [START_REF] Araujo | Endogenous Collateral[END_REF] Obviously, the plain no-short-sales restriction also served both purposes, without any need for uniform impatience. However, it was hard to see what was the institutional framework behind constraints (a) or (b).

Knowing one needs to borrow a security in order to short it, there is a natural possession constraint, the non-negativity of title collected for each security, (in the "box"). Such non-negativity contraint is quite di¤erent from the above instrumental constraints.

In such a context, the two forms of limited re-hypothecation introduced in …nite horizon (in Bottazzi, Luque and Páscoa (2011a)) readily extend to open ended horizon, where they actually also guarantee equilibrium even with in…nite lived agents and without uniform impatience requirements. These are two institutional arrangements that end up restricting the way in which a borrowed security will be lent again several times (possibly by being shorted and then lent by the buyer 2 ). One of these two forms is the provision that the security borrowed through repo cannot be fully shorted or lent, more precisely, the haircut portion of the pledged security, paid for with client money, cannot be re-used [START_REF] Araujo | Collateral Avoids Ponzi Schemes in Incomplete Markets[END_REF] .

We call this the direct limited re-hypothecation case. The other form is based on [START_REF] Araujo | Endogenous Collateral[END_REF] The …rst purpose was met as the …nite supply of asset to pledge as collateral in Asset Backed Securities (the housing stock for Mortgages) limits possible issuance. Interestingly, in the sub-prime crisis, when in 2007 Wall-street ran out of assets to pledge (sub-prime loan) to satiate the investor appetite, such a bound of the underlying stock of assets was stepped aside. The issuance went on arti…cially using the derivative markets through the synthetic CDOs market -see a narrative account of this in Michael Lewis's "The Big Short."). But when pledged assets are replaced by a combination of cash and CDS, the quantity of available asset is no upper bound on issuance. We think such arrangements should be discouraged in the future. [START_REF] Araujo | General Equilibrium, Wariness and Bubbles[END_REF] Aitken and Singh (2009) addressed re-hypothecation in the narrower sense of lending a security that one had borrowed. Bottazzi, Luque and Páscoa (2011a) include also the rehypothecation that occurs when the shorted security is lent by the buyer. [START_REF] Araujo | Collateral Avoids Ponzi Schemes in Incomplete Markets[END_REF] This is a more and more common practice, and for the equivalent of haircuts in the derivatives market (initial margins), such initial margins are segregated when cleared on an exchange. Practice and regulation are more and more pushing towards such a segregation, after the Lehman crisis.

the current market practice of bounding the positions of dealers, who are exempt from paying haircut and actually collect haircuts from customer. We call this the constrained dealers case.

Limited re-hypothecation bounds the leverage that can be done in the economy. One may think that the haircut alone is enough to bound leverage (by the inverse of the haircut rate). This is true when leverage is done by just trading in one security and its respective repo market. But, when we combine these operations on more than one security, the haircut alone cannot bound leverage:

the haircut paid in one security can be compensated by the haircut collected in another security. In this respect, our conclusion is that the institutional arrangements that bound leverage will also prevent Ponzi schemes. [START_REF] Bottazzi | Security Markets Theory: Possession, Repo and Re-Hypothecation[END_REF] The equilibrium existence result has an important implication in terms of security pricing. A central result in the literature of asset pricing bubbles (see Santos and Woodford (1997) and Magill and Quinzii (1996)) asserts that, if short sales were allowed, the (previous) no-Ponzi schemes conditions (debt constraints coupled with uniform impatience) end up ruling out also bubbles, for de ‡ators yielding …nite present value of wealth and when assets were in positive net supply.

In contrast, the two forms of limited re-hypothecation we introduced are now compatible with incomplete market bubbles for positive net supply securities, under time and state separable preferences.

We illustrate this result with two examples, one for each form of limited rehypothecation. There, consumers are impatient, although not uniformly (the discount factor is not stationary as beliefs change across di¤erent paths). This scenario allows for bubbles in positive net supply securities with associated repo market. The de ‡ator is given by the marginal rates of substitution and yields a …nite present value of wealth. This result adds to the examples by Santos and Woodford (1997) and Páscoa, Petrassi and Torres-Martinez (2011) that showed how an incomplete markets scenario without uniform impatience can generate bubbles of securities that cannot be shorted.

In the equilibria of these examples, there are no positive shadow prices for the box constraint and, therefore, the security price is just equal to the series of de ‡ated future expected cash ‡ows plus the bubble. If the box constraint had a shadow price, the security would be on special at that node. [START_REF] Bottazzi | On the 2008 dollar squeeze[END_REF] In general, specialness and the bubble are compatible, at any node. These are now the two possible causes for the security price to exceed the series of discounted expected cash ‡ows. There is an important reason why we include possession value in the fundamental value of the security: arbitrage should be de…ned by taking the box non-gnegativity constraint into account. We also show that, under the Kuhn-Tucker de ‡ator process, a bubble is, nevertheless, incompatible with an asymptotic specialness (a positive di¤erence between the repo and the general collateral rates, in the limit).

The next sections are as follows. In section 2 we present an in…nite horizon model with repo markets. of with date less or equal than T:

In…nite lived agents and commodities

There is a …nite set of in…nite lived agents I: There are L di¤erent types of commodities that agents use for consumption. Each agent i 2 I has an endowment of commodities ! i 2 R L D ++ ; which is assumed to be bounded from below at each node. Total endowments at node are denoted by

P i ! i : The consumption set is R L D + : Agent i's utility function, given by U i : R L D + ! R + [ f+1g, is
assumed to be separable in time and states of nature, i.e., for any x 2 R L D ++ , we de…ne U i (x) P 2D u i (x ); where u i (:) is the Bernoulli utility function at node . Moreover, we also impose the following standard assumptions:

(A1) (i) For any node 2 D; the function u i is twice continuously di¤erentiable and strictly increasing; (ii) u i (0) = 0 and

P 2D u i ( ) < +1; (iii) Du i (x ) 2 R L ++ ; 8x 2 R L + ; (iv) 8c 2 R; the set [u i ] 1 (c) is closed in R L ++ , and (v) at every x 2 R L ++ ; h 0 D 2 u i (x ) h < 0, 8h 6 = 0. 6 (
A2) Commodity endowments are uniformly bounded away from zero: for any i 2 I, ! i o 0.

Security markets

We conceive the initial period 0 as a situation where issuance has already happened, and issued securities have been placed. Each agent i thus has initial endowments e i 2 R J ++ of securities, describing his holdings when trading starts.

Securities are traded at every node in the event-tree. We denote a trade in security j at node 2 D by y i j : Agent i's security j position at node is ' i j : At the initial node 0 the position is ' i j 0 = e i j + y i j 0 : For node > 0 , the corresponding position is ' i j = ' i j + y i j (the previous position ' i j plus current trade y i j ). A short sale at node 2 D occurs when the position gets negative, that is, when ' i j < 0.

We model securities as real assets. The case of securities with exogenous nominal yields could be easily accommodated. The real proceeds of security j at node > 0 are given by a non-zero vector B j 2 R L + . Moreover, we assume that each good has at least some security paying in that good. [START_REF] Du¢ E | Special Repo Rates[END_REF] Formally, we assume that for any 2 D, the real returns matrix B of type L J does not have null rows. Given spot prices at node 2 D, p 2 R L + , the nominal return of security j is then given by p B j : The security market transactions of node take place at a price denoted by q j : Notice that this characterization of the returns matrix actually is in accordance with the common view of understanding securities as valuables -a security can be thought to represent the value of a commodity basket.

Taking into account security proceeds, we have that the total resources of physical commodities at node 2 D are

P i !i P i ! i + P j B j P i ' i j 2.

Repo markets

We introduce repo trading by using the variable z. Notation follows Bottazzi, Luque and Páscoa (2011). An agent is said to be long in repo, denoted by z > 0;

if he borrows the security in exchange of a cash loan. On the other hand, the agent is short in repo, denoted by z < 0, if he is the lender of the security and the borrower of cash. Repos are traded at every node 2 D. The loan associated with repo is j z j , where z j represents the amount of security j engaged in the repo and j is the haircutted price of the collateralized loan signed at node . The haircut (1 h j ) 2 [0; 1]; exogenously given in this model, is imposed to compensate the lender of funds with the risk associated with a simultaneous default and adverse market move of the security lent. [START_REF] Geanakoplos | Liquidity, Default and Crashes: Endogenous Contracts in General Equilibrium[END_REF] For the sake of simplicity and following typical market practice, we assume that all repos on the same security and same node 2 D share a common haircut. [START_REF] Hart | On the Optimality of Equilibrium when the Market Structure is Incomplete[END_REF] The collateralized loan price is then j = h j q j . We assume that, for a repo contract signed at node , the repurchase takes place at the following date (that is, at t( ) + 1) at repo interests r j . The repo rate (or interest rate) on a repo loan at node is

r j 1 j .
Securities borrowed can be re-hypothecated, that is, lent or short sold (and then lent by the buyer). Bottazzi, Luque and Páscoa (2011a) showed that, for economies with more than one security, existence of equilibrium in a two-dates economy requires the rehypothecation of securities to be bounded. Two scenarios were shown to verify this: direct limited rehypothecation and constrained dealers.

These scenarios are now adapted to our in…nite horizon model.

I) Direct limited re-hypothecation

In this case the security borrower cannot re-hypothecate a certain percentage of the borrowed amount of the security. To make a simple argument we assume that haircut paid is being segregated. [START_REF] Hernández | Competitive Equilibria for In…nite-Horizon Economies with Incomplete Markets[END_REF] Thus, the re-hypothecation percentage should be no more than 1 -the haircut rate, as the excess security pledged is not lent on. That is, the fraction H j of a security j that can be sold or lent out in repo after being borrowed at node must satisfy H j h j < 1. The haircut posted by counterparties (and paid for with their own funds) is set aside.

Budget constraints in this in…nite horizon framework depart from previous ones in the literature. Agent i's budget constraints at nodes 0 and > 0 are, respectively, g i 0 (a 0 ; p; q; r) p 0 (x 0 ! i 0 ) + q 0 ('

0 e i + h 0 z 0 ) 0 (1)
g i (a ; a ; p; q; r) p (x ! i )+q (' ' +h z ) p B ' q r h z ) 0

where a (x ; ' ; z ) denotes an agent's plan at node : 11 This budget constraint says that the value of the excess of commodities demand, asset purchases and repo long positions (z > 0) minus short positions (z < 0) must be below the commodity returns of accumulated asset positions (' ) and the repo loan repayments of the previous repo agreements (q r h z ).

Agent i's box constraint of security j at node is

f i;H j (a ) ' j + H z + j z j 0 ( 3
)
10 There is an evolution of practice in that direction, it may be accentuated by regulation, including interpretation of existing law (as haircut is paid for with customer money). Note how this assumption is not dissimilar to the segregation of initial margin by exchanges for vanilla derivatives like interest rate swaps. where z + j = maxf0; z j g and z j = minf0; z j g. So, z j = z + j z j : Now,

f i;H j is a concave function as f i;H j (a ) = ' j + z j (1 H )z + j :

II) Constrained dealers

As it currently happens, not all agents get the cash bene…t from collecting a haircut when borrowing a security -in other words, giving a cash loan and receiving a security as collateral that is worth more. Agents whose business is intermediation (Dealers/Prime brokers) have this cash bene…t, but in practice have their positions bounded in value by regulation. The mechanisms often are among other things BIS ratios limits. Their customers (e.g., hedge funds, mutual funds, retail securities brokers, private banks and insurance companies)

do not face such regulation on their positions, but must pay haircut when lending securities. We refer to the former as dealers (D) and to the latter as non-dealers (ND). As it usually occurs, we assume that non-dealers only engage in repo with dealers.

Due to the asymmetry in the haircut treatment, we have to use di¤erent variables for security borrowing and lending. We denote the long and short repo positions at node by 0 and 0, respectively. When a dealer is long in repo (and thus the non-dealer is short) the interest rate that applies is r 1 ; while if the dealer is short in repo (and thus the non-dealer is long) the interest rate that applies is r 2 : Notice that dealers like to take positions in these two variables, whereas non-dealers prefer to have just one of them non-null (see Bottazzi, Luque and Páscoa (2011a)). However, regulation prevents dealers from taking positions in the two variables that are too big.

Dealer i 2 D budget constraints at nodes 0 and > 0 are, respectively, g i 0 (a 0 ; p; q; r) p 0 (x 0 ! i 0 ) + q 0 ('

0 e i + h 0 0 0 ) 0 (4)
g i (a ; a ; p; q; r) p (x

! i ) + q (' ' + h ) p B ' q (r 1 h r 2 ) 0 (5)
Non-dealer i 2 ND budget constraints at nodes 0 and > 0 are, respectively, g i 0 (a 0 ; p; q; r) p 0 (x 0 ! i 0 ) + q 0 ('

0 e i + 0 h 0 0 ) 0 (6)
g i (a ; a ; p; q; r) p (x

! i ) + q (' ' + h ) p B ' q (r 2 r 1 h ) 0 (7)
For both dealers and non-dealers, the security j box constraint of an agent i 2

D [ ND at node is f i j (a ) ' j + j j 0 (8)
We assume that the borrowing of securities by dealers is bounded by regulation in the following way:

(A3) The real values of dealer i's long repo positions are uniformly bounded, i.e., for each j,

q j P l p l j M j (9)
As the regulator sets upper bounds on real rather than just nominal values, this policy does not su¤er from monetary illusion. That is, the regulator takes into account in ‡ation (or de ‡ation) when setting the bounds.

Notice that (A3) implies that the real value of dealers'short sales is also uniformly bounded. In fact, the box constraint together with (A3) imply q j P l p l ' j M j . Notice that the weaker version holds: nominal values of security borrowing and short-sales are clearly (uniformly) bounded (as (p ; q ) can be normalized to be in the simplex). This implies that feasible security (and repo positions) have

Individual optimality and equilibrium

We now introduce the equilibrium concept for this economy. Let us …rst denote by a i a plan for agent i at (p; q; r): For case I (direct limited rehypothecation) an individual plan is given by the vector

a i = (x i ; ' i ; z i ) 2 R L D + R J D R J D ;
where x i is subject to the sign constraint x i 0. For case II (constrained dealers) this plan is

a i = (x i ; ' i ; i ; i ) 2 R L D + R J D R J D + R J D +
; where x i 0;

i 0 and i 0. We say that a plan a i is individually admissible for agent i if it satis…es constraints (2) and (3) in case I, ( 5), ( 8) and (9) in case II with i being a dealer, and ( 7) and (8) if case II with i being a non-dealer. Consumer i's problem is said to be optimal if he chooses an admissible plan a i that maximizes his utility U i . We assume that De…nition 1: An equilibrium for this economy consists on a vector of prices

(p; q; r) 2 R L D + R J D + R J D
+ and individual plans ( a i ) i2I , such that, (i) for each agent i 2 I, the plan a i is optimal at prices (p; q; r):

(ii) at any node 2 D; commodity, security and repo markets clear, that is,

X i2I x i = 0; X i2I ( ' i e i ) = 0;
and

X i2I z i = 0, if case I X i2ND i = X i2D i = 0 and X i2D i = X i2ND i = 0, if case II
Before characterizing the Euler and transversality conditions for this economy, let us …rst denote agent i's Lagrangian at node by [START_REF] Lewis | The Big-Short[END_REF] L i (a ; a ; i ; i ; p; q; r) u i (x ) i g i (a ; a ; p; q; r)

+ i f i (a )
Let L i 1 and L i 2 denote super-gradient vectors of L i with respect to present and previous plans a and a , respectively. Proposition 1: If a i is an optimal solution to (i) of De…nition 1, then there exist multipliers ( i ) 2D 0 and ( i ) 2D 0 together with super-gradients for u i and f i at a i such that the following Euler conditions are satis…ed

L i 1 ( a i ) + X 2 + L i 2 ( a i ) 0 (10) 0 @ L i 1 ( a i ) + X 2 + L i 2 ( a i ) 1 A a i = 0 ( 11
)
and the following transversality condition holds lim sup

T !1 X :t( )=T 0 @ X 2 + L i 2 ( a i ) a i 1 A 0 (12)
Proposition 2: Let a i be an admissible plan for agent i, at prices (p; q; r).

Suppose a i satis…es Euler and transversality conditions ( 10), ( 11) and ( 12), respectively, at (p; q; r). Then, a i is individually optimal if, for any plan a i satisfying the corresponding individual's budget and box constraints [START_REF] Magill | Incomplete Markets Over An In…nite Horizon: Long Lived Securities and Speculative Bubbles[END_REF] , we have lim sup

T !1 X :t( )=T L i 1 ( a i )a i 0 (13)
Inequality (13) means that the plan a, of consumption, security and repo trades, when weighted by current marginal bene…ts, has a vanishing value. In a model where naked short-sales are naively allowed, (13) reduces to the wellknown condition that the plan a should not allow the agent to be a debtor at in…nity (lim inf T P :t( )=T q j ' j 0). Now the current marginal bene…t must take into account the cost of borrowing securities (that is, q j would be reduced by j ). See Section 4 on repo Ponzi schemes for a detailed analysis. Santos and Woodford (1997) showed that an exogenous restriction to debt is equivalent to restriction (13) if the strong assumption of uniform impatience is imposed. In this paper we do not need to impose uniform impatience, as our natural modeling of short sales for securities does not need it.

Lemma 1: The present value of wealth is …nite for the Lagrange multipliers de ‡ator process i :

The proof of Lemma 1 is established in the Appendix, as in Proposition 2 in Páscoa, Petrassi and Torres-Martínez (2011). Now, let us show that: Proposition 3: Let a i be an admissible plan for agent i, at prices (p; q; r), satisfying Euler conditions (10) and (11), in cases I or II. Then, the condition (13) holds (in both cases I or II, respectively).

We get the following existence result: The proof, given in the Appendix, uses Propositions 2 and 3.

Repo Ponzi schemes

Outside of cases I or II, Ponzi schemes can be done, envolving a new type of in…nite horizon arbitrage combining security and repo positions. Suppose budget constraints are, as in case I, described by (2) but limited rehypothecation is not imposed, so that the security j box constraint is as follows, at each node :

' j + z j 0 (14)
To simplify, we assume the economy to be deterministic, although the argument extends to stochastic economies straighforwardly. Now, given any plan (x; '; z) satisfying budget and box constraints (( 2) and ( 14), respectively), for a consumer i, we reduce the security position ' t and match this by increasing the repo position z t in the same amount at some date t, with a net gain in income (due to the haircut) and then repeat this procedure at all following dates, but possibly with di¤erent amounts in order to accommodate the changes in debt and dividends. Such variation can then be scaled up arbitrarily.

To be more precise, we consider the following variation: at date t, the repo position z t is increased by " t > 0 and the security position ' t is decreased by " t :

Box constraint ( 14) remains satis…ed at date t, but this joint operation results in a gain at date t given by q t (1 h t )" t that can be spent on extra consumption. At the following date, t + 1, the consumer can accommodate the variation in dividends and debt (net of the settlement of the repo variation) by …nding " t+1 > 0; so that z t+1 is increased by " t+1 , ' t is decreased by " t > 0 and ' t+1 is decreased by " t+1 ; while preserving the budget constraint (2) at date t + 1. That is, " t+1 must satisfy q t+1 (1 h t+1 )" t+1 (q t+1 + p t+1 B t+1 )" t + q t h t r t " t 0

Hence, we must have

" t+1 (q t+1 + p t+1 B t+1 q t h t r t )" t q t+1 (1 h t+1 )
This condition can always be satis…ed by any " t+1 > 0 when the numerator on the right hand side is negative and by " t+1 > 0 large enough if the numerator is positive, which would be the case when Euler equations on security and repo positions hold [START_REF] Magill | In…nite Horizon Incomplete Markets[END_REF] . We repeat the procedure at the following dates and obtain a vector of increments (" t ; " t+1 ; :::) that determines an increase in the utility of consumer i: By multiplying this vector by an arbitrarily large scalar > 0 we get unbounded utility gains.

Analogous Ponzi scheme could not be done in case I or II. In case I, if z t > 0, for box constraint (3) to hold, when z t increases by " t , ' t cannot decrease more than H t " t : However, as H t h t ; the combination of these variations involves a cost to the consumer, at least equal to the extra loan given in repo (q t h t " t ) minus the decrease in the security position (q t H t " t ): If z t < 0, although it would possible to decrease both z t and ' t by " t , with a resulting income gain (q t (1 h t )" t ), such arbitrage could not be scaled up by an arbitrarily high > 0; as z t would become positive.

In case II, the box constraint (8) allows for an increment " t in t accompanied by a reduction " t in ' t : If the consumer is a non-dealer, there would be no change in income at date t (as no haircut is being collected by him, see ( 7)).

However, this joint operation yields a non-positive payo¤ at the next date, given by q t r t " t (q t+1 + p t+1 B t+1 )" t . In fact, this di¤erence is non-positive, as we know that Euler conditions hold (since an optimum was shown to exist, by Proposition 3), which tell us that q t+1 + p t+1 B t+1 = i t i t+1 q t i t i t+1 and q t r t i t i t+1

q t i t i t+1
(the inequality form is due to the sign constraint on security borrowing). This cost cannot be compensated by repeating the operation at t + 1; since again an increment in t+1 accompanied by a symmetric reduction in ' t+1 would not bring an income gain (by the same argument using (7)).

If the consumer were a dealer instead, the above joint modi…cation of t and ' t would result in an income gain (1 h t )q t " t at date t (according to (5)). However, dealers have bounded positions and, therefore, such arbitrage could not be scaled up arbitrarily. This shows the importance of regulations around balance-sheet limitation and haircut segregation for dealers.

constrained) q t+1 + p t+1 B t+1 = i t i t+1 q t i t i t+1 and q t h t r t = i t i t+1 q t h t i t i t+1 :
Using the Euler conditions on security positions, recursively, we obtain the following pricing formula for security j at node

q j = fundamental security value z }| { X > i i p B j | {z } discounted dividends + X i j i | {z } specialness + 1 i lim T X :t( )=T i q | {z } bubble ( 15
)
where the sum of the two series constitutes the fundamental value and the last term is the bubble.

Proposition 4: Bubbles cannot occur under complete markets.

Observe that the absence of bubbles refers to the unique de ‡ator, given by common marginal rates of intertemporal substitution, for which the present value of wealth is …nite (see Lemma 1).

Proof of Proposition 4:

To see this, recall that the transversality condition is lim sup

T X :t( )=T L i 1 ( a i ) a i 0 (16)
In case I we write (16) as

lim sup T X :t( )=T ( Du i + i p ) x i + + ( i q i ) ' i + ( i q h i ) z i 0 Now, i ( ' i + z i ) = 0 implies 15 lim sup T X :t( )=T i (q ' i + q h z i ) 0 + lim sup T X :t( )=T i (Du i i p ) x i 0 If i = ; 8i; then lim sup T X :t( )=T q ( X i ' i + h X i z i ) X i lim sup T X :t( )=T q ( ' i + h z i ) 0
Hence, we have lim sup T P :t( )=T q e 0; implying that lim sup T P :t( )=T q 0. At any node , 0 lim T P :t( )=T q q < 1 and lim T X :t( )=T q lim sup T X :t( )=T q so lim T P :t( )=T q = 0. Now we show that bubbles cannot occur either in case II under complete markets. In fact, in this case (16) implies that lim sup

T X :t( )=T q ( X i ' i + h ( X d d X nd nd )+ + ( X nd nd X d d )) 0
Market clearing then implies lim sup T P :t( )=T q e 0 and, therefore, for any node , lim T P :t( )=T q = 0:

However, under incomplete markets the term lim T P :t( )=T q may be positive, as shown in the examples below.

Bubbles in incomplete markets: Examples

We give now two examples where bubbles occur, under a de ‡ator yielding …nite present value of wealth, for a positive net supply asset that is shorted in equilibrium. Preferences and endowments do not satisfy uniform impatience conditions, but these are not required for existence of equilibrium in cases I or II. Recall that in earlier models, equilibrium existed when short sales were either ruled out or were treated as naked short sales (although subject to constraints on the value of borrowing), but uniform impatience had to be assumed in the latter.

Example 1: This is an example of a security and repo equilibrium for an economy with two in…nite lived agents, A and B, trading one commodity and one security in sequential incomplete markets.

i s t = i s t u + i s t d , A u = (1=2 t( )+1 ) A s t and B s t u = (1 (1=2 t( )+1 )) B s t
: We denote by stud the node attained after the history of node realizations s t by going up and then down (and similarly for other pairs of branches).

Commodity endowments have a trend component equal to 1 for both agents.

Endowment shocks bene…t agent A when down is followed by up, while agent B gets also a positive shock but when up is followed by down and at 0d . More precisely, agent A's endowment is

! A = 1 + P t( ) if = du for some 2 D
and equals 1 otherwise. Agent B's endowment is

! B = 1 + P t( ) if = 0d or
= ud for some 2 D and equals 1 otherwise. The positive shocks P t( ) will be speci…ed below.

The security pays B t units of the commodity at every node occurring at date t: [START_REF] Páscoa | Harsh Default Penalties Lead to Ponzi Schemes[END_REF] Agent A is endowed with 1 unit of the security at 0 ; while agent B is not endowed.

Equilibrium prices: Let us take the security to be the numeraire and denote the commodity price by p . We will …nd equilibrium prices that depend only on the date and denote these by p t( ) . Euler conditions on consumption are satis…ed when the Lagrange budget multiplier i is given by i = i t( ) =p : Making box shadow prices equal to zero, the Euler condition on security positions holds

if i = t( )+1 i (B t( )+1 + 1=p t( )+1 ), that is, if (1=p t( )+1 ) (1= )(1=p t( ) ) = B t( )+1 :
The Euler condition on repo positions holds if the repo rate paid at nodes of date t coincides with p t B t :

We want endowment shocks to have a value 1, in units of account. Hence, P t( ) = 1=p t( ) and solves the following di¤erence equation:

P t( )+1 (1= )P t( ) = B t( )+1
Assuming B t = k t with k 2 (0; 1) and positing P 0 = 1 the solution becomes

P t = t + k k (1= ) ( t k t ) (6.1)
Equilibrium positions: Budget and box constraints at node are, respectively,

p (x i ! i ) + y i + hz i p B t( ) ' i + (1 + p B t( ) )hz i (6.
2)

' i + y i + z i 0 (6.
3) where = h if z i > 0 and 1 otherwise. At the initial node 0 we have

(x A 0 ; ' A 0 ; z A 0 ) = (! A 0 ; 1; 0) and (x B 0 ; ' B 0 ; z B 0 ) = (! B 0 ; 0; 0): At node 0d
, agent B uses his endowment shock to purchase the security with funding, that is,

y B 0d + hz B 0d = 1, which implies y B 0d = z B 0d = a;
where a 1=(1 h): At this node agent A consumes the dividends from the aggregate endowment of the security (and also at node 0u ). Hence, To guarantee that the above plans and prices constitute an equilibrium, it just remains to check that the transversality condition (3.3) holds. Using (3.2), this condition can be written as

x B 0d = ! B 0d P 1 , x A 0d = ! A 0d + P 1 + B 1 , ' A 0d = 1 a < 0 and z A 0d = a > 1: At node 0u ,
lim sup T X :t( )=T L i 1 ( a i ) a i 0 (6.4)
As the condition (3.1) for consumption holds with equality, (6.4) becomes lim sup

T X :t( )=T i ' i + hz i 0 (6.5)
Now, notice that ' i + hz i coincides with the money position in the example of Páscoa, Petrassi and Torres-Martínez (2011), whereas by (6.1), we have

i = i 1 + k k (1= ) (1 ( k) t( ) ) (6.6)
where i is the budget multiplier in that example (that is, the multiplier that would make the Euler equation in consumption hold if p t were t ), for which

lim T P :t( )=T i ' i + hz i = 0. If k < 1 then i < i and, therefore, (6.5) holds.
Bubble: In this equilibrium the security has a price bubble. In fact, at each node the security price 1 is equal to the fundamental value plus the bubble given by (1= i ) lim T P :t( )=T i . To evaluate this limit we use (6.6), noticing that

lim T (1 + k k (1= ) (1 ( k) t( ) )) > 0 for 2k < 1= and that lim T P :t( )=T i =
i > 0 (since …at money had price 1 and zero fundamental value in the example of At node 0duu ; portfolio positions are the same as in node 0du ; whereas at 0dud positions are the same as in node 0d . We have determined positions at all types of nodes. See graphs 3 and 4 below.

(1,0,0,1)

(1,0,0,1)

(-1,1,0,0) (-1,1,0,0) (a,0,a,1) (-1,1,0,0) (a,0,a,1) + Graph 3: Agent A's positions (' A ; A ; A ; ' A + A h A ): (0,0,0,0) (0,0,0,0) (2,0,1,1) (2,0,1,1) (1-a,a,0,0) (2,0,1,1) + + +
(1-a,a,0,0)

Graph 4: Agent B's positions

(' B ; B ; B ; ' B + B h B ):
It remains to show that the transversality condition (3.3) holds for both agents. For A and B, these conditions are respectively lim sup

T X :t( )=T i ' A + A h A 0 (6.9)
and lim sup

T X :t( )=T i ' B + h A B 0 (6.10)
We now notice that ' A + A h A and ' B + h A B coincide with the term

' i + hz i of condition (6.5) in Example 1. |
In the next section we discuss whether there are other non-arbitrage de ‡ators for which the above security prices are free of bubbles.

Remark on uniform impatience: The well known result that bubbles in positive net supply assets are absent under uniform impatience for de ‡ators with …nite present value of wealth (Santos and Woodford (Theorem 3.3)) extends to securities that can only be shorted by borrowing them …rst.

De…nition 2: We say that the economy satis…es uniform impatience if there exist 2 (0; 1) and for each 2 D such that, for all x 2 R LD + with x , we have U i (x( ; 0 )) > U i (x); 8i 2 I, 2 D, where x ( ; 0 ) = x for 2 DnD( );

x ( ; 0 ) = x + and x ( ; 0 ) = 0 x for > with 0 2 [ ; 1): Moreover,

9k > 0 : ! i k > 0; 8i 2 I; 2 D:
Let us show that in case I uniform impatience implies absence of price bubbles for a security in positive net supply and a de ‡ator with …nite present value of wealth.

Denote by (p; q; r; x; y; z) an equilibrium and consider the following change in consumption and …nancial plans:

x i 7 ! x i + ; x i 7 ! x i for > , (' i ; z i ) 7 ! (' i ; z i ) for
: Under uniform impatiance an appropiate choice of ( ; ) requires (1 )q (' i + h z i ) < p ; so that optimality is not contradicted.

Notice that in case I (see proof of Proposition 3) we have ' i +h z i 0. It follows that 0 (q =p )(' i + h z i ) < 1=( 1): Adding across consumers we see that q =p is uniformly bounded. If is such that P 1 t=0 P :t( )=t p ! i < 1; using again De…nition 2, we get lim t!1 P :t( )=t q = 0 as claimed.

In case II we take the equilibrium plans of non-dealers and modify them as above. Now, (' i ; i ; i ) 7 ! (' i ; i ; i );for : Again, we require ( 1)q (' i + i h i ) < p ; where ' i + i h i 0 by the box constraint.

So, it follows that (q =p ! m ) 1); for any m 2 ND:

P i2ND (' i + i h i ) < k 1 (]ND)=(
Now, by (A3) we know that (q = P l p l )

P i2D
i is uniformly bounded and, therefore, (q = P l p ) P i2D ' i is also uniformly bounded (by the box constraint). It follows that (q = P l p ) P i2D (' i i + h i ) is uniformly bounded. [START_REF] Rincón-Zapatero | Di¤erentiability of the value function without interiority assumptions[END_REF] Putting the two results together we have that (q =p ! m )(

P i2I ' i + P i2ND i P i2D i h ( P i2ND i P i2D i
)) is uniformly bounded (as P l p l =p ! m 1= inf ! m ): Hence, (q =p ! m ) 2 l 1 and we get lim t!1 P :t( )=t q = 0 as claimed for with P 1 t=0

P :t( )=t p ! m < 1:
7 Specialness and bubbles

As seen in equation ( 15), the security price at a node may exceed the series of discounted dividends for two reasons. First, there may be positive shadow prices of box constraints at this or future nodes. Second, the security may have a price at in…nity. The former raises the fundamental value above what that discounted [START_REF] Rincón-Zapatero | Di¤erentiability of the value function without interiority assumptions[END_REF] Since the uniform bounds from below and from above follow, respectively, from ' i i i and

P i2D (' i i ) = P i2ND (' i i ) < P i2ND (' i+ i ) = P i2D (' i i ):
dividends series is. The latter creates a bubble.

A positive shadow price of the box constraint at a node implies that the security is on special. This means that the respective repo rate is below the general collateral rate, which is the prevailing repo market interest rate if the borrower of funds can choose the bond to pledge. GC is generally below uncollateralized borrowing, taken here, for simplicity, to be the interest rate ( ) on a risk free one-period bond. In fact, adding this bond to the model and adapting the argument in Bottazzi, Luque and Páscoa (2011) to case I, say, the Euler conditions on repo positions of security j and on trade of that risk-free bond are, respectively,

1 = i j i j q j h j i + (1 + ) X 2 + i i ( 17
) 1 = (1 + ) X 2 + i i (18)
Condition (18) holds in equality form as we are comparing with a risk free bond that can be shorted and is itself not on special (see Du¢ e (1996), p. 494, where such comparison is also done). Hence, < if and only if i j > 0. So specialness makes the security price to be above its series of discounted dividends.

Can the bubble occur on top or just instead of this overpricing? The next result shows that it can but the occurrence of a bubble requires specialness to fade away as time goes to in…nity.

Proposition 5: If security j has a bubble, at some node , then lim

T !1 max > :t( )=T ( j j ) = 0:
The proof of Proposition 5 is in the Appendix. [START_REF] Rockafellar | Convex Analysis[END_REF] Remark to Examples 1 and 2: Are there other de ‡ators for which the equilibrium security prices found in Examples 1 and 2 are free of bubbles? 19 [START_REF] Rockafellar | Convex Analysis[END_REF] Observe that lim T !1 P :t( )=T i j ! 0 does not, by itself, imply lim T max > :t( )=T ( j j ) = 0, as we just know that lim T max

> :t( )=T 1+ 1+ 1 lim T !1 max > :t( )=T i j
h ¯T min > :t( )=T i q j , where min > :t( )=T i q j may tend to zero. [START_REF] Santos | Rational Asset Pricing Bubbles[END_REF] We thank Manuel Santos for having risen this issue. Santos and Woodford (1997, Theorem 3.1) showed that when the supremum, over all non-arbitrage de ‡ators, of the present value of wealth is …nite, there is always a de ‡ator for which bubbles are absent. Such non-arbitrage de ‡ators were de…ned by requiring just the Euler condition on asset positions to hold with equality. For the Kuhn-Tucker de ‡ator that we use in the examples, all Euler conditions hold (namely, on consumption).

Our examples were based on an example (in Páscoa, Petrassi, Torres-Martínez (2011)) of an economy with …at money that …ts in Santos and Woodford (1997) set-up, but money has a positive price due to a bubble. Clearly, such equilibrium price cannot be obtained of bubbles for another de ‡ator with non-arbitrage conditions holding with equality (as the price would be zero, since money has no dividends). Hence, by Theorem 3.1 in Santos and Woodford (1997), the supremum over all such de ‡ators of the present value of wealth had to be in…nite in that monetary example. Nevertheless, the positive price of money can be shown to be recovered free of bubbles for a de ‡ator with non-arbitrage conditions holding with strict inequality at nodes where the portfolio constraint (a no-shortsales restriction on money) was binding.

Similarly, in the examples in this paper, we can …nd a non-arbitrage de ‡ator for which the equilibrium security price is free of bubbles, but this de ‡ator makes non-arbitrage conditions hold with strict inequality at nodes where the box constraint is binding. In this way, we keep the same security prices, but the bubble is replaced by an extra term in the fundamental value consisting of the series of box shadow prices.

Let us be more precise and start by de…ning a non-arbitrage de ‡ator.

De…nition 3: In case I, we say that for prices (q; r) there are linearized arbitrage opportunities at node if 9 (' ; z ) : ' + z 0 and A(' ; z ) > 0;

where = H if z > 0, = 1 if z < 0 and 2 [H ; 1]; and A = " q q h (q + p B ) 2 + (h r q ) 2 + # Lemma 2:
In case I, there exist no arbitrage opportunities if and only if

0; 0 such that q = X 2 + (q + p B ) + q h = X 2 + r h q + This is
A d = " q h q q (q + p B ) 2 + (h r 1 q ) 2 + (r 2 q ) 2 +

#

-for a non-dealer at node if 9 (' ; ; ) : 0; 0; ' + 0 and A nd (' ; ; ) > 0; where

A nd = " q q h q (q + p B ) 2 + (r 1 q ) 2 + (h r 2 q ) 2 + # Lemma 3:
In case II, there exist no arbitrage opportunities if and only if 9 0; 0 such that, if the agent is a dealer, we have

q = X 2 + (q + p B ) + (19) q h X 2 + r 1 h q + (20) q X 2 + r 2 q + (21)
current arrangement that limits, by regulation, the positions of dealers, whom by collecting but not posting haircut would have an incentive to borrow and lend, at the same time, large amounts of securities. Such non-convexity in dealers' attitudes contrasts with their counterparties'preference for convex combinations, as haircut is posted but not collected. Counterparties (hedge funds, mutual funds, retail securities brokers, private banks and insurance companies) do not need to have positions bounded.

There are many issues that would deserve further work. As usual in the literature, the assets'positive net supply results from initial holdings at the …rst date. Issuance at other nodes of the event-tree is not being considered. An interesting step would be to model issuance and discuss its implications (both in the equity and the debt forms). Clearly, the issuance chosen should not be bounded by simple quantitative constraints (of the type that bounded borrowing in the previous literature) but one should take into account that a large issuance may decrease the security price (raise the interest paid on debt). Price taking might be questionable in that context. Repo fails or the counterparties'default in repurchasing securities were also not addressed, but there may be interesting substitution e¤ects between not honoring repo agreements and running a Ponzi scheme. Finally, more applied work could try to look at the relative importance of bubbles and specialness as two forms of overpricing securities, both shown by us to occur only in incomplete markets.

Appendix

Proof of Proposition 1: For each agent i and for each T 2 N, we de…ne , an optimization problem with …nite horizon T by truncating the utility functions in the following way U iT (x) = P 2D T ( 0 ) u i (x ) and modifying the budget and box constraints at with t( ) T 1 in the following way. For t > T no commodity, security or repo trades can be done. At t = T commodities can be traded, securities pay dividends but are no longer traded. At T 1 repo trades cannot be done (since at the following date securities have no value) and securities are traded under a plain no-short-sales restriction.

That is, for with t( ) = T 1 we require g iT (a ; a ; p; q; r) p (x ! i )+ q (' ' ) p B ' q r h z 0 together with f iT (a ; a ) ' 0.

For such that t( ) = T we require g iT (a ; a ; p; q; r) p (x

! i ) q ' p B ' 0.
The truncated Lagrangean is de…ned at each node by L iT (a ; a ; ; ; p; q; r) u i (x ) g iT (a ; a ; p; q; r) + f iT (a )

Now, the saddle point property holds for some multipliers ( iT ; iT j ) j at a solution a iT to this truncated problem [START_REF] Uzawa | The Kuhn-Tucker Theorem in Concave Programming[END_REF] , that is, for any plan (a ) 2D T ( 0 ) satisfying relevant sign constraints we have X :t( ) T L iT (a ; a ; iT ; iT ; p; q; r) U iT x iT (22)

iT g iT ( a ; a ; p; q; r) = 0; iT f iT ( a ) = 0:

Notice that U iT x iT U i ( x i ) (as the truncated problem has additional constraints), so

X :t( ) T L iT (a ; a ; iT ; iT ; p; q; r) U i ( x i ) (24)
(i) To derive Euler conditions, as iT f iT ( ' i ; z i ) = 0; we can work with the following inequality X :t( ) T L iT (a ; a ; iT ; iT ; p; q; r) U i ( x i ) + X :t( ) T iT f iT ( ' i ; z i )

The next claim is almost the desired Euler conditions as it says that iT D 1 g iT + P 2 + iT D 2 g iT is getting closer, as T ! 1; to being a super-gradient of u i + f iT (with u i restricted to the positive orthant).

Claim 1: For each with t( ) T 1 and for any action for that node b = (x ; ' ; z ) with x 0 we have u i (x ) u i ( x i ) + (f iT (' ; z ) f iT ( ' i ; z i )) ( iT D 1 g iT ( a i ; a i ; p; q; r)+ X

2 +
iT D 2 g iT ( a i ; a i ; p; q; r))(b a i ) + X 2DnD T u i ( x i ) (26) [START_REF] Uzawa | The Kuhn-Tucker Theorem in Concave Programming[END_REF] This follows by the generalized Slater constraint quali…cation (Uzawa (1958)): by making

x 0 = ! i 0 ; ' = e i and for 6 = 0 , x = ! i + B e i , so that budget constraints hold with equality and box constraints with strict inequality.

Proof of Claim 1: This follows by using (25) making, for any node ; a = a i (1 1 ( )) + b1 ( ) where 1 ( ) = 1 if = and equal to 0 otherwise, for with t( ) T 1:

Now, we let T ! 1. The sequence ( iT ; ( iT j ) j ) T is bounded for each node by Lemma 4 below. So, we can …nd a cluster point ( i ; ( i j ) j ) for the countable product topology. Denote by the indicator function of R L + de…ned by (c) = 0 for c 2 R L + and (c) = 1 otherwise. Let u i (:) + (:) ûi (:). Taking the limit in (26) we get that i D 1 g i ( a i ; a i ; p; q; r) + P

2 +
i D 2 g i ( a i ; a i ; p; q; r) is a supergradient v 0 at point a i of the function v : (x ; ' ; z ) 7 ! ûi (x i ) + f i (' ; z ), which embodies the restriction of u i to the positive orthant :

Now, k is a super-gradient of at c i¤ k(c 0 c) 0 for any c 0 0. So kc = 0

(by picking c 0 = 0 and c 0 = 2c): Applying Theorem 23.8 in Rockafellar (1970) we write v 0 as a sum of a supergradient of the unrestricted function u i (:) + f i (:)

and a supergradient k of . We get (10) from the non-negativity of k and (11) from k a i = 0.

Observe that the equality in (10) actually holds for the coordinates of variables not subject to sign constraints.

(ii) Let us now prove that the transversality condition (12) holds. We use the inequality (24) and make a = a i if t( ) t 1 and a = 0 otherwise.

Notice that g i (0; a ; p; q; r) = p ! i + D 2 g i (p; q; r) a i , for with t( ) = t; and g i (0; 0; p; q; r) = p ! i , for with t( ) t + 1, as u i ( ; 0) = 0: Then, we obtain X :t( )=t iT D 2 g i (p; q; r) a i + X :T t( )>t iT p ! i X :t( ) t u i ( x i ) Now, by (A1)(ii) the series of utilities converges for feasible plans (that is, P :t( ) t u i ( x i ) ! 0 as t ! 1). Then, using L i 2 (a ; a ) = i D 2 g i (p; q; r) , we obtain (12).

Proof of Proposition 2: Consider any plan a i satisfying the budget, box and sign constraints. Let x i its respective consumption plan. Denote by U iT the truncation of U i to the …nite horizon T . Now, we know that

U iT (x) U iT ( x i ) X :t( ) T (L i (a i ) L i ( a i )): (27)
any T :

X :t( )=t
iT D 2 g i (p T ; q T ; r T ) a i + X :T t( )>t iT p T ! i X :t( ) t u i ( x i )

and we make T ! 1 noticing that L i 2 (a ; a ) = i D 2 g i (p; q; r) . This cocludes the proof of Theorem 1.

Lemma 4: For the sequence of horizon T truncated optimization problems with a common price vector (p; q; r) the corresponding sequence of multipliers ( iT ; ( iT j ) j ) T is bounded for each node :

Proof: This follows by adapting Lemmas T1 and T4 replacing (p T ; q T ; r T ) by (p; q; r):

Proof of Proposition 5: From equations (7.1) and ( 7.2) we have that X > :t( )=T q j h j i Now, if the right hand side tends to zero, as T ! 1, we have the claimed result, as P > :t( )=T q j i 9 0 implies max

> :t( )=T 1+ 1+
! 1: To see that P > :t( )=T i j ! 0, we use inequality (24) and make a = (0; e i ; 0) for every node . Then, X :t( ) T i p (! i + B e i ) + e i X :t( ) T i j

U i ( x i )
Making T ! 1 we see that P But, in all events we get " = 0 as can be seen by the way and i were de…ned.

  [START_REF] Levine | Debt Constraints and Equilibrium in In…nite Horizon Economies with Incomplete Markets[END_REF] For notation brevety, when two vectors a = (a 1 ; :::; a N ) and b = (b 1 ; :::; b N ); with the same dimension, appear multiplied, ab; we mean the vector a b = (a 1 b 1 ; :::; a N b N ): This is the case of products h z and r h z :

  (uniformly) bounded nominal values, which is what Bottazzi, Luque and Páscoa (2011a) assumed to show existence of equilibrium in a …nite horizon economy of type II.

Theorem 1 :

 1 Equilibrium exists, in case (I) under (A.1) and in case (II) under (A.1), (A.2) and (A.3).

Graph 1 :Graph 2 :

 12 Agent A's positions(' A ; z A ; ' A + hz A ): Agent B's positions (' B ; z B ; ' B + hz B ):

Proof of Proposition 6 :

 6 It is easy to see that the Euler equation in ' is satis…ed. Then, the Euler equation in z is satis…ed if the repo rate would be decreased by " = i p t( )+1 B t( )+1 + 1 P > i h 1

  Portfolios must satisfy the box constraint (3): security purchases can be funded and short sales can be done using what has been borrowed of the security.The in…nite tree D has two branches, up ( stu (t; s t ; u)) or down ( std (t; s t ; d)), at each node . We denote by stu the node attained after the history of node realizations s t by going up (and similarly for std ).

	Preferences are t( ) i x ; for i = A; B; where 2 (0; 1) is the discount 2 (0; 1) is the probability belief at node st satisfying i given by U i (x) = P 2D factor and i s t 0 = 1;

Ponzi schemes are avoided by limited rehypothecation (case I, with h = H) and the security has a price bubble. Preferences and endowments are adapted from an example of a monetary equilibrium in Páscoa, Petrassi and Torres-Martínez (2011), but …at money (with a no-short-sales constraint) is now replaced by a security paying real dividends.

  there are no security or repo trades.At node 0du ; agent B consumes the dividends from the aggregate endowment of the security (and also at node 0dd ). That is, x A

		0dd ) =
	(a; a):
	At node	0duu agent A consumes the dividends from the aggregate endowment
	of the security (and also at node 0dud ). Security trades are zero and positions
	become (' A 0duu ; z A 0duu ) = (a; a) and (' B 0duu ; z B 0duu ) = (1 a; a): At node 0dud ;
	security trades are y A 0dud = 1 2a and y B 0dud = 2a 1; so (' A 0dud ; z A 0dud ) = (1 a; a)
	and (' B 0dud ; z B 0dud ) = (a; a): We have determined positions at all types of nodes.
	See graphs 1 and 2 below.

0du = ! A 0du P 2 and x B 0du = ! B 0du + P 2 + B 2 : By (6.2), y B 0du + hz B 0du + ha = 1: Agent B's security trade y B 0du = 1 2a implies a position ' B 0du = 1 a: Then, (6.3) holds for z B 0dU = a. By market clearing, we get ' A 0du = a and z A 0du = a: At node 0dd ; there are no security trades and positions are (' A 0dd ; z A 0dd ) = (1 a; a) and (' B 0dd ; z B

  proven by the theorem of separation of convex cones (see Araujo, Fajardo and Páscoa (2002, Theorem 1).Let us call ( ; ) a pair of non-arbitrage de ‡ator and shadow values. In Example 1, the same security prices q are obtained if the Kuhn-Tucker de ‡ator is replaced by the non-arbitrage de ‡ator i = t( ) =p t( ) P

	Proposition 6:

i together with the shadow values process = i t =p t ; where i = i 2 t( ) if the box constraint is binding at node and z i > 0; and i = 0 otherwise.

De…nition 4:

In case II, we say that for prices (q; r) there are arbitrage opportunities -for a dealer at node if 9 (' ; ; ) : 0; 0; ' + 0 and A d (' ; ; ) > 0; where

Documents de Travail du Centre d'Economie de la Sorbonne -2012.10

In particular, if dealers were not constrained they could do a Ponzi scheme through the haircut collection process.

More precisely, this is the case when the General Collateral (GC) rate would coincide with 4 Documents de Travail du Centre d'Economie de la Sorbonne -2012.10

Documents de Travail du Centre d'Economie de la Sorbonne -2012.10

Here D denotes the di¤erential operator. Items (i), (iii) and (iv) were already assumed in Bottazzi, Luque and Páscoa (2011a). Item (v) already appeared but in the weaker form of di¤erential strict quasi-concavity. These items allow us to bound the intertemporal rates of substitution from above and from below. Inada conditions would be too extreme in this model with repo markets, where a period can be very short, possibly overnight. 6 Documents de Travail du Centre d'Economie de la Sorbonne -2012.10

For example, if there is a forward term contract for each good.

For endogenous haircuts in the case of mortgages see Geanakoplos[START_REF] Magill | In…nite Horizon Incomplete Markets[END_REF] and Araujo, Fajardo and Páscoa[START_REF] Araujo | Collateral Avoids Ponzi Schemes in Incomplete Markets[END_REF]. In Fostel and Geanakoplos[START_REF] Magill | Incomplete Markets Over An In…nite Horizon: Long Lived Securities and Speculative Bubbles[END_REF] the margins on …nancial assets collateralizing money promises are also endogenous. In a recent paper, Brunnermeier and Pedersen[START_REF] Du¢ E | Special Repo Rates[END_REF] address the dependence of margins or haircuts on asset's market liquidity.

This can and should be relaxed when we focus more on credit of the trading entitiessomething we do not go into here.

This is the Lagrangian of a convex problem and we refer the reader to Rockafellar (1970). 11 Documents de Travail du Centre d'Economie de la Sorbonne -2012.10

Constraints (2) and (3) in case I,[START_REF] Bottazzi | On the 2008 dollar squeeze[END_REF],[START_REF] Geanakoplos | Liquidity, Default and Crashes: Endogenous Contracts in General Equilibrium[END_REF] and[START_REF] Hart | On the Optimality of Equilibrium when the Market Structure is Incomplete[END_REF] in case II with i being a dealer, and[START_REF] Du¢ E | Special Repo Rates[END_REF] and (8) if case II with i being a non-dealer.12 Documents de Travail du Centre d'Economie de la Sorbonne -2012.10

These conditions hold in the form of the following equations (as the variables are not sign 13 Documents de Travail du Centre d'Economie de la Sorbonne -2012.10

Recall that lim sup(A B) lim sup(A) lim sup(B):

Notice that there is no possible confusion between the agent B and the dividends B t , as the latter are always indexed by the time subscript, while the former not.

Páscoa, Petrassi and Torres-Martínez (2011)). Notice that the Lagrange de ‡ator process ( i ) for which the bubble occurs yields a …nite present value of wealth (as already mentioned in Section 3). | Example 2: Let us now consider a di¤erent scenario, where agent A is a non-dealer and agent B is a dealer. This corresponds to case II of the analysis above. In order to characterize equilibrium in this framework, we need to do the following modi…cations with respect to Example 1 above. Now the Euler condition on repo positions holds if both repo rates (r 1 1 and r 2 1) paid at nodes of date t( ) coincides with p t( ) B t( ) : Then, agents A and B' budget constraint are, respectively,

Agents A and B'box constraints are, respectively,

' B + B B 0 (6.10)

The equilibrium positions are now the following. At the initial node 0 there are no securities nor repo trades, so An analogous proposition to Proposition 6 for case II holds with i = i 2 t( ) if the box constraint is binding at node and z i > 0; and i = 0 otherwise.

Finally, notice that the supremum of the present value of aggregate endowments, over all de ‡ators for which non-arbitrage conditions hold with equality, is still in…nite as in the example in Páscoa, Petrassi and Torres-Martínez (2011).

In fact, the way that supremum is computed (indicated by Proposition 2.2 in Santos and Woodford (1997)) in that example shows that the supremum for the economy with …at money is the same as the supremum in the economy with repo markets. This suggests that when we look for a de ‡ator for which there is no bubble, we should …nd it within the class where non-arbitrage conditions may hold with strict inequalities at some nodes.

Concluding remarks

Once we explicitly take into account the way securities are actually shorted, by borrowing them …rst in repo markets, we see that mechanisms that bound leverage and ensure equilibrium in …nite horizon will also prevent in…nite lived agents from doing Ponzi schemes. This result dispenses any uniform impatience assumptions. In this security and repo context, we see reappearing the main insight in Santos and Woodford (1996): bubbles in positive net supply securities (for de ‡ators with …nite present value of wealth) cannot occur when markets are complete but may occur in incomplete markets when consumers are not uniformly impatient. However, that room for bubbles seemed to be quite narrow before, as, in the absence of uniform impatience, short sales apparently had to be ruled out (as in the examples by Santos and Woodford (1996) or Páscoa, Petrassi and Torres-Martínez (2011)). Now, if shorting and security borrowing are properly coupled, Ponzi schemes are not always ruled out (as our example shows) but are absent when there is limited leverage. So, bubbles may occur in incomplete markets, when uniform impatience fails but leverage is adequately bounded. We presented two ways that limit the re-hypothecation of the security and the resulting leverage. One is the increasing practise (after Lehman's banruptcy) of not re-using (shorting or lending) the haircut collected when borrowing a security. The other is the Notice that by the superdi¤erential property,

Taking T ! 1 and using Euler and transversality conditions ( 10), ( 11) and ( 12), we get lim sup

and, therefore,

Proof of Lemma 1: We take inequality (24) and make a = (0; 0; 0) for every node . Then,

Making T ! 1 we see that

Proof of Proposition 3: The Euler condition on x 0 (which holds at this cluster point of …nite horizon equilibria) is

all 2 D: Therefore, (13) can be rewritten as shown next.

In case I: Let A i i q (' +h z )+ i (' +H z + z ): Then (13) reduces to requiring lim sup

to hold for any ('; z) satisfying budget and box constraints. Notice that ' +

where = H if z > 0 and 1 otherwise.

The Euler condition on ' implies that i q + i = P

Hence A i 0.

In case II: For any ('; ; ) satisfying budget and box constraints, with ( ; ) 0, and jq j ' i j j M j if i 2 D; the analogous dealer's the condition is lim sup T !1 P :t( )=T B i 0; where B i i q (' +h )+ i (' + );

and the corresponding non-dealer's condition is lim sup T !1

where

Let us start with the non-dealer's condition. We have C i ( i q + i )(' j +

) since ( i q h i ) ( i q i ) : Now, i q + i < 0 (again this follows from the Euler condition in ' ); so C i 0 and the desired condition holds.

Let us now look at the dealers'condition. Notice that i + i j

i r 1j h j (from dealer's Euler condition in j ). Then, B i P j ( i + i j q j )q j (' j j ) P j ( i i j q j )q j j . Now, for the dealers, the real value of long repo positions is uniformly bounded, that is, q j j = P l p l is uniformly bounded. As i h j i j q j 0, it su¢ ces to show that P :t( )=t P l p l 2 l 1 . We know that P 2D p ! i < 1. Since ! i o 0, we have P :t( )=t i P l p l 2 l 1 , as desired.

Proof of Theorem 1: An equilibrium for a …nite horizon T economy is de…ned by requiring market clearing and individual optimality subject to these modi…ed constraints. We denote it by (p T ; q T ; r T ; a T ) and ( iT ; iT ) be an associated vector of Lagrange multipliers of budget and box constraints (respectively) of agent i. Notice that p T 0 (by monotonicity), we can normalize (p T ; q T ) in the simplex.

Lemma T1: For each node we have iT P l p T l U i ( )=! i , where ! i = min l ! i l .

Proof: By the saddle point property, for any (a ) 2D T ( 0 ) satisfying relevant sign constraints, we have X :t( ) T L i (a ; a ; iT ; iT ; p T ; q T ; r T ) U iT ( x iT ) (30)

iT g iT ( a ; a ; p T ; q T ; r T ) = 0; iT f iT ( a ) = 0:

Now let a be the plan consisting of consumption ! i , null security position and null repo positions for nodes such that t( ) t 1 while being zero at other nodes. Then the above saddle point inequality implies

) and we get the result.

Lemma T2: P l p T l 0 is bounded away from zero.

Proof: We consider the following plan for case I:

where E l 0 is the canonical vector in the direction of commodity l at node 0 : This plan satis…es the date 0 budget constraint

Observe that it is enough to make b min l;j f! i l 0 ; e i j g; as (p T 0 ; q T 0 ) belongs to the simplex. Subsequent budgent constraints are clearly satis…ed too. This plan clearly satis…es box constraints. Now, if p T l 0 ! 0 then (1 p T l 0 )x iT would converge to x iT in the l 1 norm topology, for which the utility is continuous. Thus, U iT (x T ) > U iT (x iT ) for T large enough, a contradiction. The proof for case II is analogous.

Lemma T3: For each node we have the sequence ( iT ) T bounded.

Proof: Lemmas T1 and T2 imply that ( iT 0 ) T is bounded. To see what happens at the immediately following nodes, we recall that the …rst order condition on ' j for the truncated economy is

iT (q T j + p T B j ) + iT j Take = 0 and suppose that iT ! 1 for some 2 + 0 . Then, q T j should go to 0 for every j: But, as (p T ; q T ) is in the simplex, is would follow that P l p T l ! 1: Then, by Lemma T1, iT 9 1; a contradiction. We have shown that ( iT ) T bounded for every such that t( ) = t( 0 ) + 1: The proof for the following nodes proceeds in the same way using the above …rst order condition.

Lemma T4: For each node we have the sequence ( iT j ) T bounded.

Proof: Recall that in case I the …rst order condition on z j for the truncated economy is iT q T j h j = X 2 + iT q T j h j r j + iT j j

where j = H j if z iT j > 0 and j = 1 otherwise. So, iT j iT q T j h j =H j and the result follows by Lemma T3.

In case II the …rst order condition on j for the truncated economy requires

For both types of agents, we have iT j iT q T j ; and Lemma T3 applies again.

We already know that as (p T ; q T ) is in the simplex, the sequence (p T ; q T ) T is bounded, node by node. For the price variable r T that was left outside the simplex, we say the following:

Lemma T5: The sequence (r T ) T is bounded for each :

Proof: We show that R T 1=r T is bounded away from 0. In case I, the …rst order condition on z j implies that

So, if (for a subsequence) R T j ! 0, then iT = iT ! 0; for any 2 + : Now, by (A1) we have that

) is bounded away from zero, it follows that p l p l ! 0; so p l ! 0 (as p l is bounded). This is impossible for = 0 ; by Lemma T2.

For 6 = 0 the following argument applies. Since ; where the left hand side is bounded (by A1), if p l ! 0, then iT 0 iT ! 0: Now, iT is bounded, so iT 0 ! 0; implying D l u i 0 (x iT 0 ) ! 0: However, lim inf T D l u i 0 (x iT 0 ) 2 D l u i 0 ([0; 0 ]); which by (A1) cannot contain 0: The proof for case II is analogous, using the …rst order condition on j : Hence, the above Lemmas allow us to …nd for (p T ; q T ; r T ; a T ; ( iT ; iT ) i; ) a cluster point (p; q; r; a; ( i ; i ) i; ), as T ! 1 (for the countable product topology). We claim that (p; q; r; a) is an equilibrium.

Notice that Euler conditions hold at (p; q; r; a; ( i ; i ) i; ); by taking limits on …rst order conditions of …nite horizon economies. Then, by Propositions 1, 2 and 3 it su¢ ces to show that the transversality condition is satis…ed. This follows as in item (ii) of the proof of Proposition 1 since the following inequality holds for