BANKRUPTCY IN A MODEL OF UNSECURED CLAIMS

Aloisio P. Araujo

IMPA and EPGE/FGV (Rio de Janeiro)

Estrada Dona Castorina, 110 22460-320 Rio de Janeiro, Brasil

Mário R. Páscoa Faculdade de Economia Universidade Nova de Lisboa 1070 Lisboa, Portugal

1

1Pascoa acknowledges support from FEDER and Praxis XXI (project 2/2.1/ECO 28/94).

Summary

We study a two periods model of incomplete markets with nominal assets unsecured by collateral, where agents can go bankrupt but there are no bankruptcy penalties entering directly in the utility function. We address two cases: first, a proportional reimbursement rule under bounded short sales and limited liability and, secondly, a nonproportional reimbursement rule, favoring smaller claims, without bounds on short-sales, but assuming that liability approaches total garnishment as debt goes to infinity.

INTRODUCTION

In the past ten years, a few papers have studied default in the context of the general equilibrium incomplete markets model. The pioneering work was done by Dubey-Geanakoplos-Shubik (1989) in a two-period model where defaulters suffer utility penalties and anticipate that asset returns are discounted according to the average repayment rate. In this model, default occurs when the marginal utility of income, in a particular state of nature, exceeds the marginal penalty of defaulting in a specific asset. Zame (1993) discussed efficiency properties of a single-good, countable states, version of this model and Araujo-Monteiro-Pascoa (1996, 1998) extended the model to an infinite-tree with a continuum of states at each node. However, the current decreasing deterrence role of penalties tends to question the approach. More recently, collateral was brought into the default model, by Dubey-Geanakoplos-Zame (1995). When assets are backed by collateral and utility penalties are absent, adverse selection problems are immediately ruled out, since all lenders know what to expect to receive in each state of nature (and this is the minimum between the value of the claim and the value of the exogenously fixed collateral). There is even no need to suppose that lenders deal indirectly with borrowers, through a pool of mortgages, in order to rationalize the hypothesis that returns are discounted using the average repayment rate. However, when collateral becomes endogenously chosen by borrowers this pooling step may be required again (see Araujo-Orrillo-Pascoa (1998)).

In the present paper we addresss bankruptcy, instead of default, in the context also of the general equilibrium incomplete markets model. By bankruptcy we mean a situation where an agent has no means to pay back his debt, or more precisely, that his garnishable wealth and income do not cover his debt. Bankruptcy is a very important institutional arrangement that offers protection to agents on the basis of the concept of limited liability.

When agents are liable only up to some fraction of their wealth and income, it is in their interest to go bankrupt when the debt exceeds that fraction of their estates. Creditors will then be reimbursed using the garnished estates. This procedure should not be confused with default, which is plainly a situation where debtors fail to honor their commitments, even when they could afford to do it.

If all claims were secured by collateral, bankruptcy would become uninteresting, since it could never offer the debtor a protection putting at risk the reposession of the collateral by the lender. There are, however, many promisses not backed by collateral. To simplify the analysis we will concentrate actually on a model where all claims are unsecured. A more complex version could be worked out, taking into account the priority of secured claims in the partition of the garnished estate of a bankrupt agent (or, possibly, only after tax authorities and employees were reimbursed). We assume also that bankrupt agents do not suffer any penalties entering directly in the utility function. Legal punishments are no longer contemplated in the bankruptcy laws of many countries, although reputation punishments, related to future access to credit, are clearly relevant and could still be represented in the form of subjective utility penalties, which should be taken into account when filing for bankruptcy. However, we do not attempt to model these asymmetric information aspects of the problem.

We develop a two-periods model where uncertainty may affect endowments and preferences in the second period, through the realization of S states of nature. Assets have nominal payoffs and are not secured by collateral. For each state of nature, an agent goes bankrupt when his financial surplus is negative and its absolute value exceeds the value of the garnishable endowments. We assume that some liability rule has already determined the fraction of the endowments that creditors can garnish in case of bankruptcy. This structure for second-period budget constraints introduces a nonconvexity in the consumer’s budget set of bundles and portfolios (actually, in the latter). This difficulty is overcome by considering a continuum of consumers and appealing to Liapunov’s theorem.

Our portrait of bankruptcy was contemplated in Araujo-Pascoa (1994) and has some similarities with the one used by Modica-Rustichini-Tallon (1995) in the context of an interesting temporary equilibrium model, but it differs from the latter since we assume limited liability and do not restrict bankruptcy only to unforeseen states (to which the consumer attaches zero subjective probability). We model bankruptcy as a predictable and less dramatic contingency: it may be in the consumer’s interest to pick a portfolio that provides interesting transfers of wealth across periods and states, in spite of the known risk of leading to bankruptcy and the resulting partial confiscation of estates in some states of nature.

In the first part of the paper (section 2) we assume proportional reimbursement to creditors. That is, the garnished estates of bankrupt agents are partitioned among lenders proportionally to the claims involved. Since we model transactions anonymously, assuming that lenders deal with a pool of borrowers, the above proportional rule implies that, in a rational expectations equilibrium, lenders should discount future returns using the average repayment rate (the ratio of effective payments to promissed payments). We show that, under bounded short-sales, a bankruptcy equilibrium will exist. A recent paper by Sabarwal (1999), that we received while revising our work, addresses existence of equilibria in a multiperiod bankruptcy model, under proportional reimbursement and credit constraints which make the access to credit harder the greater was the bankruptcy in the past. Even in the absence of past bankruptcy, Sabarwal’s (1999) assumes still an upper bound on credit and, therefore, his existence result depends as ours on the bounded short sales set-up.

Bounded short-sales is a convenient but arguable assumption, avoidable when assets are nominal, both in the no-default model and in the default model with penalties or collateral. However, in our model, as in the real assets no-default model, the rank of the returns matrix depends on endogenous variables, which are the repayment rates, taking into account bankruptcy by other agents. Besides, the nonconvexity of the budget set prevents us from establishing that consumer demand is differentiable (since the Jacobian of first order conditons can not be shown to be nonsingular). This difficulty seems to rule out any attempt to use differential topology approaches in the existence argument.

In the second part of the paper (section 3) we discuss implications of deviations from proportional reimbursement to creditors. There are frequent real life deviations from the proportional rule and some are actually contemplated by law in the form of seniority criteria (giving priority to tax authorities, employees, secured creditors and some among the unsecured creditors). Theoretical work has also questioned the interest of proportional rules. The bargaining approach by a Aumann-Maschler (1985) suggested equal division up to the minimum of the smallest claim and the estate (due to equal bargaining positions when the estate is smaller than any claim) and a more complex rule, according to equaldivision of contested additional claims, in general (coinciding with the nucleolus of the cooperative game played by the creditors).

We examine a reimursement rule that tends to favor smaller claims, by making the reimbursement rate decrease with the size of the claim. This rule is actually a parametric rule satisfying certain axioms introduced in the cooperative games literature on bankruptcy (see Moulin (1987) and Young (1987, 1988)). The resulting strict concavity of positive returns, contrasted with the linearity of negative returns, forces the financial surplus to go to −∞, as budget-feasible portfolios become unbounded along many of the possible directions (those which do not exhibit linear positive returns of non-defaulting assets). On the other hand, we assume, in this part of the paper, that agents are required to provide increasing liability as the debt goes to infinity. Such a liability rule should be seen as a mechanism intended to discourage irresponsible borrowers. This assumption, together with Inada’s condition an utility, makes financial surplus bounded from below by some negative amount. The existence argument is carried out by taking a sequence of equilibria of truncated economies and appealing to a multi-dimensional version of Fatou’s lemma which requires the sequence of allocations to be uniformly bounded (or at least uniformly integrable). The above reimbursement rule, together with the liability assumption, imply non-arbitrage conditions that allow us to uniformly bound portfolios of assets subject to default.

In this second part of the paper, the nonlinearity of returns requires nonlinear pricing for assets. The portfolio cost includes, in addition to the usual linear component, a spread which is a discounted expectation of the future net default (i.e., the difference between default given and default suffered), for some endogenously determined, in equilibrium, measure on states. That is, agents more prone to go bankrupt end up selling assets at lower prices (i.e., borrow at higher interest rates). Also, lenders receive discounts when buying assets due to future expected default (and these discounts are nonlinear, due to the above nonlinear returns scheme). This structure for the spread plays a crucial role in guaranteeing that, when markets clear, default is also correctly anticipated (i.e., net aggregate default is zero).

We close the paper with a short section discussing the constrained efficiency properties of equilibria.

2. PROPORTIONAL REIMBURSEMENT

2.1 The model

We consider a two period general equilibrium model. In the first period A nominal assets and L physical goods are traded. In the second period, there are S states of nature which may affect asset returns and the consumers’ endowments and preferences over L physical goods.

The continuum of consumers is modelled as the Lebesgue measure space (I, ℑ, λ) of the unit interval I ≡ [0, 1]. Preferences are time and state separable and given by a utility function u^{h}: ℝ_{+}^{L(S+1)} → ℝ of the form u^{h}(x^{h}) = ∑_{s=0}^{S} u_{s}^{h}(x^{h}) for h ∈ I. Endowments are vectors ω^{h} ∈ ℝ_{++}^{L(S+1)} , for h ∈ I. We allow for bankruptcy: agents will not pay by their entire debt if this exceeds the value of the alienable endowment. That is, consumers have limited liability in the sense that, in case of bankruptcy, only same share γ_{sℓ} ∈ [0, 1] of the endowment of good ℓ in state s may be confiscated by the creditors. If the total nominal return is lower then − ∑_{ℓ=1}^{L} p_{sℓ} γ_{sℓ} ω_{sℓ}^{h}, at some spot price vector p_{s} ∈ ℝ_{+}^{L}, then consumer h will decide to pay the latter instead.

Default by others is anticipated according to a proportional reimbursement rule. Denote consumer h’s portfolio by y^{h} ∈ ℝ and assume that the returns matrix has nonnegative elements r_{sa}. Let θ_{sa} be the mean reimbursement rate, to be formally defined below. Discounted returns G_{sa}(r_{sa} y_{a}) are given by

G_{sa}(r_{sa} y_{a}) = \{ \begin{matrix}r_{sa} y_{a} ,  if y_{a} < 0 \\ θ_{sa}r_{sa}y_{a} ,  otherwise \end{matrix}

We can now write down the budget constraints of consumer h, given asset price vector q ∈ ℝ_{+}^{A} and spot price vector p_{s} ∈ ℝ_{+}^{L}. Consumption plan x^{h} ∈ ℝ_{+}^{SL} and portfolio y^{h} ∈ ℝ^{A} must satisfy the following first period budget constraint

(1)  p_{0} · (x_{0}^{h} − w_{0}^{h}) + q · y^{h} = 0.

In each state s of the second period the budget constraint is

(2)  p_{s} · (x_{s}^{h} − ω_{s}^{h}) = max ( ∑_{a=1} G_{sa}(r_{sa} y_{a}), − ∑_{ℓ=1}^{L} p_{sℓ} γ_{sℓ} ω_{sℓ}^{h}) ≡ f_{s}^{h}

Fig. 1

Figure 1 illustrates the shape of the function f_{s}.

When consumer h goes bankrupt, the total amount available to partially reimburse creditors will be ∑_{ℓ} p_{sℓ} γ_{sℓ} ω_{sℓ}^{h}+∑_{a}(G_{sa}^{h})^{+} where z^{+} stands for max{0, z}. This amount will be allocated across assets sold by agent h according to the weights τ_{sa}^{h} = r_{sa}(y_{a}^{h})^{−} / ∑_{b} r_{sa}(y_{b}^{h})^{−} where z^{−} stands for min{0, z}.

Default is correctly anticipated in each asset market provided that

∫_{B_{s}} ( ∑_{ℓ} p_{sℓ}γ_{sℓ}ω_{sℓ}^{h} + ∑_{b}(G_{sb}^{h})^{+})τ_{sa}^{h} dλ(h) (3)

− r_{sa} ∫_{B′_{s}}(y_{a}^{h})^{−} dλ(h) = ∫_{I}(G_{sa}^{h})^{+} dλ(h)

where B_{s} = {h: f_{s}^{h} = − ∑_{ℓ} p_{sℓ} γ_{sℓ} ω_{sℓ}^{h}}.

The above equation states that the aggregate payments, by defaulters and nondefaulters, related to asset a, should match aggregate revenues due to this asset, in any state s.

Notice that the left hand side of equation (3) can be rewritten as r_{sa} ∫_{I}(y_{a}^{h})^{+} d(λ(h) − ∫_{I}(f_{s}^{h} − ∑_{b} G_{sb}^{h})τ_{sa}^{h} dλ(h). The mean reimbursement ratio is defined by (3) above, since the right hand side is θ_{sa} ∫_{I}(y_{a}^{h})^{+} dλ(h), or equivalenty by:

(3′)  θ_{sa} = 1 − \frac{∫_{I}(f_{s}^{h} − ∑_{b} G_{sb}^{h})τ_{sa}^{h} dλ(h)}{r_{sa} ∫_{I}(y_{a}^{h})^{+} dλ(h)} ·

There is no need to contemplate the possibility of selling short and buying a same asset, since this action would be dominated (G_{sa}(0) > −r_{sa}y_{a} + θ_{sa}r_{sa}y_{a}), for θ_{sa} < 1).

An equilibrium is a vector (x, y, p, q, θ) ∈ L^{1}(I, ℝ_{+}^{(S+1)L} × ℝ^{A}) × ℝ_{+}^{(S+1)L+A} × [0, 1]^{SA} such that

• (x̄^{h}, ȳ^{h}) maximizes u^{h} subject to (1) and (2), for a.e. h ∈ I

• markets clear, i.e. ∫_{I}(x^{h} − ω^{h})dλ(h) = 0 and ∫_{I} y^{h} dλ(h) = 0

• (3) holds, for any (s, a).

2.2 The result

We say that an allocation of bundles x ∈ L^{1}(I, ℝ_{+}^{(S+1)L}) is uniformly bounded from above, or from below, if there exist M, m > 0 such that ess sup x_{sl} < M for any s = 0, …, S, l = 1, …, L, or if ess inf x_{sl} > m for any s = 0, …, S, l = 1, …, L, respectively.

Theorem 1. Suppose the endowment allocation w is uniformly bounded, from above and from below, and asset short-sales are required to be bounded from below, by t ∈ ℝ_{++}^{A}, say. Then, an equilibrium exists.

To prove this theorem, consider a sequence of economies E^{n} with truncated consumption sets L_{n} = [0, n]^{L(S+1)} and truncated portfolio sets K_{n} = [−d, n]^{A}. Define, for each n, a generalized game played by the continuum of consumers and S + 1 + SA additional players.

(0) Each consumer h chooses (x^{h}, y^{h}) ∈ L_{n} × K_{n} in order to maximize his utility subject also to the budget constraints (1) and (2), given (q, p, θ).

The additional players are as follows.

(i) S auctioneers choose p_{s} ∈ Δ^{L−1} in order to maximize p_{s} · ∫_{I}(x_{s}^{k} − ω_{s}^{h})dλ(h)

(ii) one auctioneer chooses (p_{0}, q) ∈ Δ^{L+A−1} in order to maximize

p_{0} · ∫_{I}(x_{0}^{h} − ω_{0}^{h})dλ(h) + q · ∫_{I} y^{h}dλ(h)

(iii) SA additional agents choose θ_{sa} ∈ [0, 1] according to (3’) above (which is equivalent to (3)).

Lemma 1. A pure strategies noncooperative equilibrium of the generalized game of a truncated economy is an equilibrium for this economy, for n sufficiently large.

We say that a vector (x, y, p, q, θ) ∈ L^{1}(I, ℝ_{+}^{(S+1)L} ×ℝ^{A})×ℝ_{+}^{(S+1)L+A} ×[0, 1]^{SA} is a pure strategies noncooperative equilibrium for the generalized game of the truncated economy E^{n} if (0), (i), (ii) and (iii) are satisfied.

Proof: At an equilibrium of the game, the optimality conditions of consumers’ problems and of problems (i) and (ii) imply that

∫_{I}(x_{0}^{h} − ω_{0}^{h})dλ ≤ 0, ∫_{I} y^{h} dλ ≤ 0,

and, therefore, by (iii), aggregating (3’) over assets, we have

∫_{I} f^{h} dλ(h) = ∫_{I} ∑_{a} G_{sa}^{h} dλ(h) + ∑_{a}(1 − θ_{sa})r_{sa} ∫_{I}(y_{a}^{h})^{+} dλ(h) = R_{s} ∫_{I} y^{h} dλ(h)

then,

p_{s} ∫_{I}(x_{0}^{h} − ω_{0}^{h})dλ(h) = ∫_{I} f^{h} dλ(h) ≤ R_{s} ∫_{I} y^{h} dλ ≤ 0.

Now, p_{sℓ} > 0, for any s = 0, 1, …, s and any ℓ = 1, …, h, for n sufficiently large. Otherwise, every consumer would choose x_{sℓ}^{h} = n, implying that ∫_{I} x_{sℓ}^{h} dλ = n > ∫_{I} w_{sℓ}^{h} dλ, for n > ess sup w_{sℓ}. But when p_{oℓ} > 0 we must have ∫_{I}(x_{oℓ}^{h} − w_{oℓ}^{h})dλ(h) = 0, since ∑_{ℓ=1}^{L} p_{oℓ}(x_{oℓ} − w_{oℓ}^{h})dλ(h) + ∑_{j=1} q_{j} ∫_{I} y_{j}^{h} dλ(h) is a null sum of non positive terms, hence a sum of null terms.

Moreover, q_{j} > 0, for any j = 1, …, A. Otherwise, every consumer would choose y_{j} = n, implying ∫_{I} y_{j}^{h} dλ(h) = n > 0. But when q_{j} > 0 we must have ∫_{I} y_{j}^{h} dλ(h) = 0, by the argument above.

Finally, since ∫_{I} f_{s}^{h} dλ(h) = R_{s} ∫_{I} y_{s}^{h} dλ(h) and the latter is zero, it follows that p_{s} · ∫_{I}(x_{s}^{h} − w_{s}^{h})dλ(h) = ∫_{I} f_{s}^{h} dλ(h) = 0 and therefore ∫_{I}(x_{s}^{h} − w_{s}^{h})dλ(h) = 0, for n sufficiently large. In fact, we showed that p_{s} ≫ 0 and therefore excess supply is ruled out in any spot market, since ∑_{ℓ=1}^{L} p_{sℓ} ∫_{I}(x_{sℓ}^{h} − w_{sℓ}^{h})dλ(h) is a null sum of h nonpositive terms, hence a sum of L null terms.

To show that the game has an equilibrium we have to solve the difficulties created by the nonconvexity of the budget set.

Lemma 2. The generalized game of a truncated economy has a noncooperative equilibrium, in pure strategies.

Proof: First, let us extend the game to allow for consumers’ mixed strategies in portfolios. Let ρ^{h} be a regular probability measure on the Borel sets of K_{n}. We replace budget constraints (1) and (2) by the extended budget constraints

∫_{K_{n}} q · y^{h}dρ^{h}(y^{h}) = p_{0} · (w_{0}^{h} − x_{0}^{h})
p_{s} · (x_{s}^{h} − ω_{s}^{h}) = ∫_{K_{n}} f_{s}^{h} dρ^{h}(y^{h})

We also replace, in problem (ii), ∫_{I} y^{h} dλ(h) by ∫_{I} ∫_{K_{n}}(id)dρ^{h}(y^{h})dλ(h) and ∫_{I} f_{s}^{h} dλ(h) by ∫_{I} ∫_{K_{n}} f_{s}^{h} dρ^{h}(y^{h})dλ(h).

Let us replace, in problem (iii) θ_{sa} by

1 − ∫_{I} ∫_{K_{n}} (f_{s}^{h} − ∑_{b} G_{sb}^{h})τ_{sa}^{h} dρ^{h}(y^{h})dλ/r_{sa} ∫_{I} ∫_{[−n,n]}(id)^{+} dρ_{a}^{h}(y_{a}^{h})dλ(h)

Moreover, let us replace, in problems (i) and (ii), ∫_{I}(x_{s}^{h} −w_{s}^{h})dλ(h) (s = 0, 1, …, s) by ∫_{I} ∫_{K_{n}}(d_{s}^{h}(y^{h})−w^{h})dρ^{h}(y^{h})dλ(h), where d^{h}(y^{h}) = argmax{u^{h} : x^{h} satisfies the constraints (1) and (2), given the portfolio y^{h}}. In this framework, consumers choose pairs (x^{h}, ρ^{h}) of bundles and mixed strategies over portfolios and other agents choose (p, q, θ) given the consumers’ profile of mixed strategies over portfolios. Even though one might argue that an equilibrium of this extended game is not a mixed strategies equilibrium, all that matters for our purpose is the fact that a degenerate equilibrium of the extended game is an equilibrium of the original game.

The extended game has an equilibrium, possibly in mixed strategies over portfolios. In fact, the best response correspondence of agents (i), (ii) and (iii) is upper semicontinuous on the profile of consumers’ probability measures on K_{n} (with respect to the weak^{∗} topology on the dual of L^{1}(I, C(K_{n}))) and the profiles set is compact (for the same topology).

Now, it is possible to show the existence of a pure strategies equilibrium. First, notice that the functions in problems (i), (ii) and (iii) depend on the profile of mixed strategies (ρ^{h})_{h} only through finitely many m indicators of the form ∫_{I} ∫_{K_{n}} z^{h}(y^{h})dρ^{h}(y^{h})dλ(h), where z_{j} ∈ L^{1}(I, C(K_{n})), for j = 1, …, m.

Secondly, let B^{h} be the mixed strategies (in portfolios) best response correspondence of consumer h, given by ρ ∈ B^{h}(p, q, θ) if there exists x ∈ [0, n]^{L(S+1)} such that (x, ρ) maximizes u^{h} subject to the extended budget constraints, given (p, q, θ). Now

∫_{I} ∫_{K_{n}} z^{h}(y^{h})dB^{h}(y^{h})dλ(h) = ∫_{I} ∫_{K_{n}} z^{h}(y^{h})d ext B^{h}(y^{h})dλ(h),

for z ≡ (z_{1}, …, z_{m}) ∈ L^{1}(I, (C(K_{n}))^{m}), since

con ∫_{K_{n}} z^{h}(y^{h})d ext B^{h}(y^{h}) = ∫_{K_{n}} z^{h}(y^{h})dB^{h}(y^{h}) ⊆ ℝ^{m}

and theorem I.D.4 in Hildenbrand (1974) applies.

Then, given a mixed strategies (in portfolios) equilibrium profile (ρ^{h})_{h}, there exists a portfolio profile (y^{h})_{h} such that the Dirac measure at y^{h} is an extreme point of B^{h} (evaluated at the equilibrium levels of the variables chosen by the atomic players) and (y^{h})_{h} can replace (ρ^{h})_{h} and keep all equilibrium conditions satisfied (without changing the equilibrium levels of the variables chosen by the atomic players but replacing the former equilibrium bundles by d^{h}(y^{h})).

Therefore, each truncated economy has an equilibrium (x, y, p, q, θ).

Let Z_{n} = (x_{n}, y_{n}, p_{n}, q_{n}, θ_{n}) be an equilibrium sequence for E^{n} and let y_{jn}^{+h} = max(0, y_{jn}^{h}) , y_{jn}^{−h} = max(0, −y_{jn}^{h}).

Fatou’s lemma, part I (on functions mapping into the positive orthant of a finitedimensional space, see appendix), allows us to infer from the convergence of ∫_{I}(x_{n}^{h}, y_{n}^{+h}, y_{n}^{−h})dλ (passing to a subsequence if necessary, since the sequence is bounded by (∫_{I} w^{h} dλ, t, t)) that there exists an integrable function (x, y^{+}, y^{−}) mapping into ℝ_{+}^{L(S+1)+2A} such that ∫_{I}(x^{h}, y^{+h}, y^{−h})dλ = lim ∫_{I}(x_{n}^{h}, y_{n}^{+h}, y_{n}^{−h})dλ and, for a.e. (x^{h}, y^{+h}, y^{−h}) is a cluster point of (x_{n}^{h}, y_{n}^{+h}, y_{n}^{−h}).

Claim: (x^{h}, y^{h}) ≡ (x^{h}, y^{+h} − y^{−h}) is an optimal solution of consumer h’s problem at the cluster point (p, q, θ).

Suppose (x̃^{h}, ỹ^{h}) satisfies the budget constraints at (p, q, θ) and u^{h}(x̃^{h}) > u^{h}(x^{h}). Lower hemi-continuity of the budget correspondence (by lower hemi-continuity of its interior and result 4 in Hildenbrand (1974), page 26) implies that there is a sequence (x̃^{hn}, ỹ^{hn}) converging to (x̃^{h}, ỹ^{h}) such that (x̃^{hn}, ỹ^{hn}) satisfies the budget constraints at (p^{n}, q^{n}, θ^{n}). Then u^{h}(x̃^{hn}) > u^{h}(x^{hn}), for n sufficiently large. Moreover, the quantity constraints of the truncated economies are satisfied, since (x̃^{hn}, ỹ^{hn}) converges. This would contradict optimality of (x^{hn}, y^{hn}).

Then the cluster point (p, q, θ) must satisfy (p, q) ≫ 0, which is a necessary optimality condition for the consumers’ problems, in the untruncated economy (where consumption and asset purchases are not bounded). Hence, ∫_{I}(x^{h} − w^{h})dλ(h) = 0 and ∫_{I} y^{h} = 0, since the sequences of allocations (x_{n}^{h}, (y_{n}^{h})^{+}) became uniformly bounded (and part II of Fatou’s

lemma applies): x_{noℓ}^{h} ≤ (ess sup_{h,ℓ} w_{oℓ}^{h}/p_{0ℓ}) + t ∑_{j} q_{j} and (y_{na}^{h})^{+} ≤ (ess sup_{h,ℓ} w_{0ℓ}^{h}/q_{a}) + t ∑_{j} q_{j}.

This completes the proof of Theorem 1.

2.3 An example

Consider a two-agents (or a continuum set with two types), two-states economy with a single physical good and a single asset whose returns vector is (1,1) and suppose v = 1 is the short sales constraint level. Preferences of agents i = 1, 2 are given by u^{1} = 4ℓn x_{0} + ℓn x_{1} + ℓn x_{2} and u^{2} = 2ℓn x_{0} + 3ℓn x_{1} + 3ℓn x_{2} ; endowments of agent 1 are ω1 ¯0 = 1, ω_{1}^{1} = 2, ω_{2}^{1} = 2 and of agent 2 are ω_{0}^{2} = 2, ω_{1}^{2} = 1, ω_{2}^{2} = 1. Let γ = 1/4 be the limited liability coefficient. We will construct an equilibrium by setting portfolios as follows: y^{1} = −1, y^{2} = 1. Let p_{s} = 1 for s = 0, 1, 2. Then, x_{s}^{1} = 2+max{−1, −1/2} = 3/2, s = 1, 2, whereas x_{1}^{2} = 1 + θ_{1} and x_{2}^{2} = 1 + θ_{2} . Market clearing in the spot markets implies θ_{1} = θ_{2} = 1/2.

The first period budget constraint is x_{0}−φ_{0}+πy = 0 and, therefore, x_{0}^{1} = 1+π whereas x_{0}^{2} = 2 − π. The market clearing equation for the first period commodity is satisfied for any value of π. The two equations (3’) governing the repayment rates are also satisfied. It remains to check the individual optimality of the above portfolio choices. The problem of consumer 1 reduces to choosing y ≥ −1 in order to maximize the reduced utility function 4ℓn(1−y)+2ℓn(2+max{δy, −1/2}), where δ = 1 if y < 0 and δ = θ = 1/2 otherwise. This reduced utility function is decreasing in y (it is continuous and piecewise differentiable with negative derivative) and therefore y = −1 is the optimal choice. The reduced problem of consumer 2 is to choose y ≥ −1 in order to maximize 2ℓn(2−y)+6ℓn(1+max{δy, −1/4}), which is increasing for y ∈ (−1/4, 1) and decreasing for y ∈ (−1, −1/4) ∪ (1, 2), tending to −∞ as y approaches 2. Consumer 2’s utility at y = 1 is higher than at y = −1 and therefore y = 1 is the optimal choice.

3. NONPROPORTIONAL REIMBURSEMENT

3.1 The model

We assume now that reimbursement ratios decrease with the size of the claim. This nonproportional reimbursement scheme will be crucial for our result on existence of equilibria which dispenses with bounded short-sales. Discounted returns are given by G_{sa}(r_{sa} y_{a}). The function G_{sa} coincides with the identity map on (−∞, 0] and is strictly concave on [0, +∞) when other agents default on asset a in state s. More precisely, let θ_{sa} be the mean reimbursement rate. For θ_{sa} = 1, G_{sa} is still the identity map on [0, +∞), but, for θ_{sa} < 1, the derivative of G_{sa} is equal to 1 at the origin and tends to zero as y_{a} → +∞. In section 3.7 we allow for a broad class of nonproportional reimbursement rules G_{sa} satisfying certain axioms, but to be more explicit, let us start by assuming that the precise form of the function G_{sa} is endogenously determined within the family of functions of the form G_{sa}(z) = g_{sa}(z)z where the reimbursement ratio is given by g_{sa}(z) = 1 − exp(−δz^{−γ}). The parameters δ and γ are related to the inflection point (c, d) of the curve g by γ = [1 − ℓn(1 − d)]^{−1} and δ = c^{γ}(1 − γ)/γ. Now d = θ_{sa} and c is to be determined endogenously. Notice that θ_{sa} → 1 implies g_{sa} → 1 and that dG/dc > 0, d^{2}G/dc^{2} < 0, for θ_{sa} < 1 (see appendix).

Fig. 2  Fig. 3

We assume also that the liability coefficients γ_{sℓ} tend to one as the debt goes to −∞. More precisely, suppose there are minimal liability coefficients γ̄_{sℓ} ∈ [0, 1] and that the actual liability ratios are given by γ_{sℓ} = γ̄_{sℓ} + (1 − γ̄_{sℓ})φ(∑_{a} G_{sa}), where φ: ℝ → [0, 1] is such that φ(z) = 0 for z ≥ − ∑_{ℓ} p_{sℓ} γ̄_{sℓ} ω_{sℓ}^{h} and φ(z) → 1 as z → −∞. This specification for the liability structure together with Inada’s condition on utility will imply the existence a lower bound on the debt. Figure 4 illustrates the shape of the function f_{s}.

Fig. 4

The second-period budget constraints are given by (2) as before, where G_{sa} has this new specification.

In the first period, we add to the linear cost of the portfolio q · y^{h} a spread which is a discounted expected value of the difference between the default given by agent h and the default suffered by this agent. Formally, the first period constraint is written as

(1′)  q · y^{h} + t ∑_{s=1}^{S} η_{s}(f_{s}^{h} − R_{s}y^{h}) + p_{0} · (x_{0}^{h} − ω_{0}^{h}) = 0.

Here f_{s}^{h} − R_{s}y^{h} = f_{s}^{h} − ∑_{a} G_{sa} − (R_{s}y^{h} − ∑_{a} G_{sa}) is the difference between default given by agent h in state s and default suffered by this agent in this state. The discount factor t ∈ ℝ_{+} and the probability measure η ∈ Δ^{S−1} are to be determined endogenously.

In equilibrium, (t, η) will be such that aggregate net default is nonpositive, that is, the mean of the map h ↦ (f_{s}^{h} − R_{s}y) is nonpositive, for each state s. To ensure that, for each asset default is correctly anticipated, requires, in addition, the endogeneous adjustment of the abcisses c_{sa} of the inflection point of the reimbrusement schedule g_{sa}.

Equation (3), requiring default to be correctly anticipated, still holds. Now, the mean reimbursement ratio θ_{sa} is defined by

θ_{sa} = \frac{∫_{I}(G_{sa}^{h})^{+} dλ(h)}{r_{sa} ∫_{I}(y_{a}^{h})^{+} dλ(h)} ·

and, therefore, (3’) still holds also.

We did not contemplate the possibility of buying and setting short in a same asset, since this action would be dominated in terms of second period returns (due) to the strict concavity of G_{sj}) and would lead to no gains in the first period either, assuming that the negative part of the spread (that is, the compensation for default suffered, given by t ∑_{s} η_{s} (∑_{j} G_{sj} − R_{s}y)) is calculated on the basis of the net holdings of each asset.

We can now define an equilibrium. Given an economy (ω, R, γ̄) ∈ L^{1}(I, ℝ_{+}^{(S+1)L}) × ℝ_{+}^{SA} × [0, 1]^{SL}, a vector (x̄, ȳ, p, q, t, η, c, θ) ∈ L^{1}(I, ℝ_{+}^{(S+1)L}) × L^{1}(I, ℝ^{A}) × ℝ_{+}^{(S+1)L} × ℝ^{A} × ℝ_{+} × Δ^{S−1} × ℝ_{+}^{2SA} is an equilibrium if

• (x̄^{h}, ȳ^{h}) maximizes u^{h} subject to (1’) and (2), for a.e. h ∈ I

• markets clear, i.e. ∫_{I}(x^{h} − ω^{h})dλ(h) = 0 and ∫_{I} y^{h} dλ(h) = 0

• (3) holds, for any (s, a).

3.2 The result

Theorem 2. Suppose

(A1) u_{s}^{h} is C^{1} on ℝ_{++}^{L} and ||∇u_{s}^{h}(x_{s}^{h})|| → ∞ as x_{s}^{h} → 0

(A2) {u^{h}}_{h} and {Du^{h}}_{h} are equicontinuous

(A3) the endowment allocation ω ∈ L(I, ℝ^{L(S+1)}) is continuous and uniformly bounded from above and from below

(A4) for (^{n}x^{h},^{n}y^{h}) satisfying the budget constraints, at (p^{n}, q^{n}, θ^{n}, t^{n}, η^{n}, c^{n}), we have

||∇u_{s}^{h}(^{n}x_{s}^{h})|| (f_{s}^{h})′(∑_{j} G_{sj}(r_{sj}^{n}y_{j}^{h})) → +∞ when ∑_{j} G_{sj}(r_{sj}^{n}y^{h}) → −∞ (and there-

fore ^{n}x_{s}^{h} → 0), for any h

Then, an equilibrium exists.

Assumptions (A2) and (A3) introduce compactness on the set of consumers’ characteristics. Assumption (A4) says that, as the debt becomes unbounded, in some state, and therefore, consumer’s income and consumption tend to zero (since the liability coefficients approach one), the marginal utility of income will go to +∞ faster than the derivative of the alienable endowment (with respect to the debt) tends to zero. We knew already that marginal utility of income would go to +∞ as income approaches zero, as the endowment of a defaulter is eventually totally confiscated. We knew also that this endowment is being confiscated at a decreasing rate, as the debt increases. We now assume the former dominates the latter, so that the indirect marginal utility, with respect to portfolio returns, explodes as the debt becomes unbounded.

Let us give an example of a liability rule and an utility function satisfying assumption (A4). To simplify assume there is a single physical good. The liability coefficients γ_{s} are given by \overline{γ_{s}} + (1 − \overline{γ_{s}})φ(∑_{a} G_{sa}) where φ(z) = 1 − exp\{−(z + \overline{γ_{s}}φ_{s})^{2}\} for z ≤ −\overline{γ_{s}}φ_{s} and φ(z) = 0 otherwise. Notice, φ(z) → 1 as z → −∞, that is, asymptotically, we have total garnishment, as the debt goes to infinity. To see that (A4) holds, notice that φ′(z) = 2(z + \overline{γ_{s}}ω_{s}) exp\{−(z + \overline{γ_{s}}ω_{s})^{2}\} which goes to zero as z → −∞. Take a logarithmic utility, u_{s}(x) = ℓn x, then u′_{s}f′_{s} → ∞ as ∑_{a} G_{sa} → −∞. In fact, for

∑_{a} G_{sa} < −\overline{γ_{s}}ω_{s} we have f_{s} = −γ_{s}ω_{s}, f′_{s} = −(1 − \overline{γ_{s}})φ′(∑_{a} G_{sa})ω_{s} and x_{s} = (1 − γ_{s})ω_{s},

then u′_{s}(x_{s})f′_{s}(∑_{a} G_{sa}) = −(1 − \overline{γ_{s}})φ′(∑_{a} G_{sa})ω_{s}u′_{s}((1 − γ_{s})ω_{s}). Hence, u′_{s}(x_{s})f′_{s}(∑_{a} G_{sa}) =

−\frac{φ′(∑_{a} G_{sa})}{1 − φ′(∑_{a} G_{sa})} = −2(∑_{a} G_{sa} + \overline{γ_{s}}φ_{s}) which goes to ∞ as ∑_{a} G_{sa} tends to −∞.

The following four sections present the proof of the theorem.

3.3 Equilibria of truncated economies

As in the proof of Theorem 1 above, consider a sequence of truncated economies D^{n}. Consumption sets are L_{n} = [0, n]^{L(S+1)}, as before, but portfolio sets are now K_{n} = [−n, n]. For each n, the generalized game is now played by the continuum of consumers, the S + 1 + SA fictitions agents contemplated earlier (choosing prices and mean reimbursement rates) and, now, also SA additional agents choosing c_{sa} = [0, r_{sa}n] in order to adjust the reimbursement schedules. These last SA players maximize min(0, ∫_{I}(G_{sa}^{h}(r_{sa}y_{a}, c_{sa}))^{+} dλ(h)−θ_{sa}r_{sa} ∫_{I}(y_{a}^{h})^{+} dλ(h)). These last SA players are refered to as the schedulers.

Lemma 1 can be redone to show that a pure strategies equilibrium of the generalized game is an equilibrium for the truncated economy, for n sufficiently large. Notice that, at a strategic equilibrium, ∫_{I}(x_{0}^{h} − w_{0}^{h})dλ ≤ 0, ∫_{I} y^{h} dλ ≤ 0 and ∫_{I}(f_{s}^{h} − R_{s}y^{h})dλ ≤ 0 (by the optimality of the strategy of the first period auctioneer and Walras’ law). Then, p_{s} · ∫_{I}(x_{s}^{h} − w_{s}^{h})dλ = ∫_{I} f_{s}^{h} dλ ≤ R_{s} ∫_{I} y^{h} dλ ≤ 0.

Notice that at, an optimum of the problem of the SA schedulers, the objective function always takes value zero and ∫_{I}(G_{sa}^{h})^{+} dλ − θ_{sa}r_{sa} ∫_{I}(y_{a}^{h})^{+} dλ ≥ 0. Since we knew already that ∫_{I}(f_{s}^{h} − R_{s}y^{h})dλ ≤ 0 it follows that equation (3) holds, for any (s, a), and ∫_{I} f_{s}^{h} dλ(h) = R_{s} ∫_{I} y^{h} dλ(h).

In fact, ∫_{I}(f_{s}^{h} − ∑_{j} G_{sj}^{h})τ_{sa}^{h} dλ = (1 − θ_{sa})r_{sa} ∫_{I}(y_{a}^{h})^{+} dλ, by definition of θ_{sa}. Now

∫_{I}(G_{sa}^{h})^{+} dλ ≥ θ_{sa}r_{sa} ∫_{I}(y_{a}^{h})^{+} dλ implies (I) ∫_{I}(f_{s}^{h} − ∑_{j} G_{sj}^{h})τ_{sa}^{h} dλ ≥ r_{sa} ∫_{I}(y_{a}^{h})^{+} dλ −

∫_{I}(G_{sa}^{h})^{+} dλ, for any (s, a). On the other hand, ∫_{I}(f_{s}^{h} − R_{s}y^{h})dλ ≤ 0 is equivalent to (II) ∫_{I}(f_{s}^{h} − ∑_{j} G_{sj}^{h})dλ ≤ ∫_{I}(R_{s}y^{h} − ∑_{j} G_{sj}^{h})dλ. Now (II) implies that (I) must hold as an equality, for any a, since ∑_{a} τ_{sa}^{h} = 1.

As in the proof of Lemma 1, p ≫ 0 and q ≫ 0 for n large enough, implying that commodity and asset markets will not be in excess supply either.

Lemma 2 can also be redone to show the existence of a price strategia equilibrium. The extended game, over mixed strategies, is as in Lemma 2, except for the extended first period budget constraint and the extended problems of the SA schedulers. The former

is now ∫_{K_{n}}[q · y^{h} + ∑_{s} β_{s}(f_{s}^{h} − R_{s}y^{h})]dρ^{h}(y^{h}) = p_{0} · (w_{0}^{h} − x_{0}^{h}), where ρ^{h} is, as before, a

probability measure on Borelians of K_{n}. In the latter, ∫_{I}(y_{a}^{h})^{+} dλ should be replaced by ∫_{I} ∫_{−n,n]}(id)^{+} dρ_{a}^{h} dλ and ∫_{I}(G_{sa}^{h})^{+} dλ by ∫_{I} ∫_{−n,n]}(G_{sa}^{h})^{+} dρ_{a}^{h} dλ, where ρ_{a}^{h} is the marginal on coordinate a. The argument follows as in Lemma 2, and therefore, each truncated economy has an equilibrium (x_{n}, y_{n}, p_{n}, q_{n}, t_{n}, η_{n}, θ_{n}), where t_{n} = ∑_{s} β_{s}^{n} and η_{n} = β^{n}/t_{n}.

3.4 A default barrier

Before discussing the properties of the sequence of truncated equilibria, as n → ∞, we need to show that, at each vector of parameters of a consumer’s problem, the debt will be bounded, and then use this fact to derive a set of nonarbitrage conditions, which will be satisfied by the equilibrium sequence, beyond a certain order.

Lemma 3. For each vector (p, q, θ, t, η, c) and each state s, →_{h} ess inf, ∑_{j} G_{sj}^{h} > −∞ on the set of portfolios which are admissible and undominated in consumers’ problem.

Proof: Let us show first that for each consumer h there is a lower bound on ∑_{j} G_{sj}^{h} and then we will argue that this bound is uniform on consumers. Suppose that for some consumer there is a sequence y_{n} of portfolios which satisfies the budget constraints and ||y_{n}|| → +∞. Looking first at the subsets J^{+} and J^{−} of assets bought and sold which diverge at the fastest rate, we assume that the set S_{1} = {s : ∑_{j∈J^{+}} r_{sj}δ_{sj} − ∑_{j∈J^{−}} r_{sj} < 0} is nonempty, for δ_{sj} = 1 when θ_{sj} = 1 and δ_{sj} = 0 otherwise. If S_{1} were empty, for any pair (J^{+}, J^{−}), we were done. Consider also S_{2} = {s : ∑_{j∈J^{+}} r_{sj}δ_{sj} − ∑_{j∈J^{−}} r_{sj} = 0} and

S_{3} = {s : ∑_{j∈J^{+}} r_{sj}δ_{sj} − ∑_{j∈J^{−}} r_{sj} > 0}. Notice that for s ∈ S_{1} , ∑_{j} G_{sj}^{h} → −∞ and f′_{s} → 0,

whereas for s ∈ S_{3} , ∑_{j} G_{sj} → +∞ and f′_{s} → 1. For s ∈ S_{2} , if there exists j ∈ J^{+} such that θ_{sj} < 1 for some s ∈ S_{2} , then ∑_{j} G_{sj} → +∞ and f′_{s} → 1 in this state; otherwise, ∑_{s} G_{sj} stays bounded in S_{2} .

Consider now the problem of utility maximization over bundles, at the financial income derived from a given portfolio. That is, max{u^{h} : x^{h} satisfies (1) and (2), given y^{h}}. The indirect utility function v^{h} is differentiable on portfolios. We claim that ∑_{j∈J^{+}} \frac{∂v}{∂y_{j}} −

∑_{j∈J^{−}} \frac{∂v}{∂y_{j}} goes to −∞, when the pair (J^{+}, J^{−}) is such that S_{1} ≠ ∅. This would imply that such a sequence y_{n} would never be chosen. Let us evaluate this directional derivative, denoting by λ_{s} (s = 0, 1, …, s) the multipliers of the constraints and by r_{s} the difference

∑_{j∈J^{+}} r_{sj}δ_{sj} − ∑_{j∈J^{−}} r_{sj}.

∑_{j∈J^{+}} \frac{∂v}{∂y_{j}} − ∑_{j∈J^{−}} \frac{∂v}{∂y_{j}} = ∑_{s} λ_{s}f′_{s}(r_{s} + ∑_{\begin{matrix}j∈J^{+} \\ θ_{sj}<1 \end{matrix}} G′_{sj}) −

λ_{0}[ ∑_{j∈J^{+}} q_{j} − ∑_{j∈J^{−}} q_{j} + t ∑_{s} η_{s}f′_{s}(r_{s} + ∑_{\begin{matrix}j∈J^{+} \\ θ_{sj}<1 \end{matrix}} G′_{sj}) − t ∑_{s} η_{s}( ∑_{j∈J^{+}} r_{sj} − ∑_{j∈J^{−}} r_{sj})].

Now the expression inside squared brackets tends to

∑_{j∈J^{+}} q_{j} − ∑_{j∈J^{−}} q_{j} − t ∑_{s∈S_{2}∪S_{3}} η_{s} ∑_{j∈J^{+}:θ_{sj}<1} r_{sj} − t ∑_{s∈S_{1}} η_{s}( ∑_{j∈J^{+}} r_{sj} − ∑_{j∈J^{−}} r_{sj}) ≡ N.

If N < 0, then q y + t ∑_{s} η_{s}(f_{s} − R_{s}y) would go to −∞ and x_{0} to +∞, implying that λ_{0} = ||∇u_{0}||/||p_{0}|| would stay bounded (possibly converge to zero). That is, N negative is incompatible with λ_{0} becoming unbounded (as a result of x_{0} tending to zero).

Now, let us look at the second period terms. For s ∈ S_{3} , r_{s}+ ∑_{\begin{matrix}j∈J^{+} \\ θ_{sj}<1 \end{matrix}} G′_{sj} tends to r_{s} > 0,

but λ_{s}f′_{s} stays bounded (since ∑_{j} G_{sj} → +∞ and x_{s} → +∞). For s ∈ S_{1}, r_{s} + ∑_{\begin{matrix}j∈J^{+} \\ θ_{sj}<1 \end{matrix}} G′_{sj} tends to r_{s} < 0 and λ_{s}f′_{s} → +∞, by assumption (A4), as

∑_{j} G_{sj} → −∞, f_{s} → −p_{s}w_{s} and x_{s} → 0.

For s ∈ S_{2} , r_{s} + ∑_{\begin{matrix}j∈J^{+} \\ θ_{sj}<1 \end{matrix}} G′_{sj} → r_{s} = 0. Now, if (a) slower diverging assets determine

that ∑_{j} G_{sj} → +∞ (or if there is j ∈ J^{+} such that θ_{sj} < 1), then λ_{s}f′_{s} stays bounded (as in S_{3}). When (b) slower diverging assets determine that ∑_{j} G_{sj} → −∞, we have λ_{s} → ∞ and f′_{s} → 0. If in these states there are no fastest bought assets for which θ_{sj} < 1, the term λ_{s}f′_{s}[r_{s} + ∑_{\begin{matrix}j∈J^{+} \\ θ_{sj}<1 \end{matrix}} G′_{sj}] will be zero throughout the sequence. Otherwise, ∑_{j} G_{sj} might

tend to −∞ in these states (when the slower assets follow a sequence n and the faster ones follow n^{α}, with α < 1/(1 − γ)), but at lower rates than in states belonging to S_{1} .

Hence, even if λ_{s}f′_{s} ∑_{\begin{matrix}j∈J^{+} \\ θ_{sj}<1 \end{matrix}} G′_{sj} would tend to +∞ for some states in S_{2} , we could still claim

that these terms would be dominated by ∑_{s∈S_{1}} λ_{s}f′_{s}r_{s}, which tends to −∞. Finally, when (c) ∑_{j} G_{sj} stays bounded, λ_{s}f′_{s} will stay bounded also and λ_{s}f′_{s} ∑_{\begin{matrix}j∈J^{+} \\ θ_{sj}<1 \end{matrix}} G′_{sj} → 0.

Then ∑_{j∈J^{+}} \frac{∂v}{∂y_{j}} − ∑_{j∈J^{−}} \frac{∂v}{∂y_{j}} → −∞ and this implies that the sequence y_{n} leads to unbounded utility losses, even if there were other assets, outside of J^{+} ∪ J^{−}, diverging at lower rates.

The equicontinuity assumption on utility and marginal utility, together with the continuity assumption on endowments, imply that the lower bound on ∑_{j} G_{sj}^{h} is uniform on consumers. In fact, let L^{h} = inf ∑_{j} G_{sj}^{h} and pick L̃^{h} < L^{h}, for each h. Now, for each h, there exists ε^{h} > 0, such that for h′ ∈ (h − ε^{h}, h + ε^{h}) we still have ∑_{j} G_{sj}^{h′} > L̃^{h}. Passing now to a finite subcover {(h − ε^{h}, h + ε^{h})}_{h=1}^{H} we find a uniform lower bound on ∑_{j} G_{sj}^{h}.

Remark 1: Suppose (θ^{n}) is such that r_{s}(θ^{n}) ≡ ∑_{J^{+}} r_{sj}δ_{sj}^{n} − ∑_{J^{−}} r_{sj} < 0, for some state s and some pair (J^{+}, J^{−}) of subsets of assets, but at θ = lim_{n}θ^{n} we get r_{s}(θ) = 0 (that is, r_{s} suffers a positive jump at the limit point of θ^{n}). Let y^{hn} be consumer h’s choice at (p^{n}, q^{n}, θ^{n}, t^{n}, η^{n}, c^{n}). Then, we can still claim that the sequence ∑_{j} G_{sj}(y_{j}^{hn}) is uniformly

bounded from below. If it were not bounded, then ∑_{J^{+}} \frac{∂v^{h}}{∂y_{j}} − ∑_{J^{−}} \frac{∂v^{h}}{∂y_{j}} would tend to −∞. The uniformity of the bound follows from equicontinuity of utility and marginal utility, together with continuity of endowments, as in the proof of the lemma above.

3.5 Nonarbitrage conditions

Let us examine now the conditions that asset prices q_{a}, mean reinbursement rates θ_{sa} and spread coefficients (t, η) must satisfy to rule out the possibility of arbitrage. Given y ∈ ℝ^{A}, let f_{s}(y) = t ∑_{s=1}^{s} η_{s}(R_{s}y−f_{s}(y))−q·y. In the context of our model, by arbitrage we mean the existence of a sequence (y_{n}) ⊆ ℝ^{A} of portfolios, satisfying the budget constraints, such that f_{s}(y_{n}) → +∞, for at least one s (s = 0, 1, …, s), and f_{s}(y_{n}) stays bounded for other states. Let δ be the S · A matrix whose elements are given by δ_{sj} = 1, when θ_{sj} = 1, and δsj = 0, otherwise.

Lemma 4: q ∈ ℝ^{A}, θ ∈ [0, 1]^{SA}, t ∈ ℝ_{+} and η ∈ Δ^{S−1} do not allow for arbitrage only if

(i) q_{j} ≥ t ∑_{s:θ_{sj}<1} η_{s}r_{sj}, when θ_{j} ≠ 1.

(ii) q_{j} > 0, when θ_{j} = 1

(iii) for each pair (J^{+}, J^{−}) of subsets of assets, such that θ_{j} ≠ 1 for some j ∈ J^{+}, when

( ∑_{j∈J^{+}} r_{sj}δ_{sj} − ∑_{j∈J^{−}} r_{sj})_{s=1}^{S} ≥ 0, we have ∑_{j∈J^{+}} (q_{j} − t ∑_{s:θ_{sj}<1} η_{s}r_{sj}) − ∑_{j∈J^{−}} q_{j} ≥ 0.

(iv) for each pair (J^{+}, J^{−}) of subseets of assets, such that θ_{j} = 1, ∀ j ∈ J^{+}, when

( ∑_{j∈J^{+}} r_{sj}δ_{sj} − ∑_{j∈J^{−}} r_{sj})_{s=1}^{S} ≥ 0, we have ∑_{j∈J^{+}} q_{j} − ∑_{j∈J^{−}} q_{j} > 0.

Proof: The first period budget constraint can be written as q · y + t ∑_{s} η_{s}(f_{s} − ∑_{j} G_{sj}) −

t ∑_{j:y_{j}>0} ∑_{s:θ_{sj}>0}(r_{sj}y_{j} − G_{sj}) = p_{0} · (w_{0} − x_{0}). By lemma 3, f_{s} − ∑_{j} G_{sj} is bounded, ∀ s.

Now, suppose (i) does not hold and let y_{j} → +∞. Then, f_{s} → +∞ for some s, (q_{j} −

t ∑_{s:θ_{sj}<1} η_{s}r_{sj})y_{j} → −∞ and G_{sj} → +∞ (s = 1, …, S). By L’Hopital’s rule, lim f_{0}(y) =

[ lim (q_{j} − t ∑_{s:θ_{sj}<1} η_{s}r_{sj})y_{j}] {1 + lim [t ∑_{s:θ_{sj}<1} η_{s}G′_{sj}/(q_{j} − t ∑_{s:θ_{sj}<1} η_{s}r_{sj})]}

= +∞, since G′_{sj} → 0 in states where θ_{sj} < 1. Then, (q, θ, t, η) would allow for arbitrage. We have shown (ii) earlier (in the proof of lemma 1).

Suppose now that (iii) does not hold and let y_{j}^{n} = n for j ∈ J^{+} and y_{j}^{n} = −n for j ∈ J^{−}.

When n → +∞, lim_{n} f_{s}(y^{n}) = +∞, for same s ∈ {1, …, s} where ∑_{j∈J^{+}} r_{sj}δ_{sj}−∑_{j∈J^{−}} r_{sj} >

0 or θ_{sj} ≠ 1 for some j ∈ J^{+} (since ∑_{j} G_{sj} would go to +∞, even if the linear terms would

cancel out). Now, lim_{n} f_{0}(y^{n}) = −[lim_{n} ( ∑_{j∈J^{+}} (q_{j} − t ∑_{s:θ_{sj}<1} η_{s}r_{sj})n − ∑_{j∈J^{−}} q_{j}n)]{1 +

lim_{n} [t ∑_{j∈J^{+}} ∑_{s:θ_{sj}<1} η_{s}G′_{sj}/( ∑_{j∈J^{+}} (q_{j} − t ∑_{s:θ_{sj}<1} η_{s}r_{sj}) − ∑_{j∈J^{−}} q_{j})]} = +∞ and arbitrage is

allowed.

If (iii) does not hold, it must be the case that ∑_{j∈J^{+}} r_{sj} − ∑_{j∈J^{−}} r_{sj} > 0, for some state s = s̄. Otherwise, the rank of R could not be A. Now, let y_{j}^{n} = n, for j ∈ J^{+}, and

y_{j} = −n, for j ∈ J^{−}. When n → ∞, fs̄ → +∞. If ∑_{j∈J^{+}} q_{j} − ∑_{j∈J^{−}} q_{j} < 0, then

lim_{n} f_{0}(y^{n}) = −lim_{n} ( ∑_{j∈J^{+}} q_{j} − ∑_{j∈J^{−}} q_{j})n = +∞ and if ∑_{j∈J^{+}} q_{j} = ∑_{j∈J^{−}} q_{j}, f_{0}(y^{n}) would

stay bounded. In either case, arbitrage is allowed.

Notice that, by lemma 3, the pairs (J^{+}, J^{−}) for which ∑_{j∈J^{−}} r_{sj}δ_{sj} − ∑_{j∈J^{−}} r_{sj} < 0 in some state are of no interest.

3.6 Asymptotic properties of truncated equilibria

If n is sufficiently large, the equilibrium values of (q, θ, t, η) in a truncated economy E^{n} are nonarbitrage vectors. Otherwise, every consumer would choose x_{sℓ}^{h} = n (for some s = 0, 1, …, s and some ℓ = 1, …, L) and feasibility would fail for n sufficiently large.

Now, (p_{0}^{n}, q^{n}, β^{n}) ⊆ Δ^{L+A+S−1}, where β^{n} = t^{n}η^{n}, (p_{s}^{n}) ⊆ Δ^{L−1} (s = 1, …, s) and θ^{n} ∈ [0, 1]^{SA}. Then these sequences have cluster points (p_{0}, q, β), (p_{s})_{s=1}^{s} and θ. We would like to infer that (q, θ, t, η) is of nonarbitrage (for t = ∑_{s} β_{s}, η = β/t).

Lemma 5: A cluster point (q, θ, t, η) of the equilibrium sequence (q^{n}, θ^{n}, t^{n}, η^{n}) of the truncated economies is still a nonarbitrage vector.

Proof: Suppose not. Then either (a) there is a pair (J^{+}, J^{−}) such that ∑_{J^{+}} r_{sj}δ_{sj}^{n} − ∑_{J^{−}} r_{sj} is negative for some state along the sequence but becomes zero at the cluster point, which might not satisfy ∑_{J^{+}} (q_{j} − t ∑_{s:θ_{sj}<1} η_{s}r_{sj}) ≥ ∑_{J^{−}} q_{j}, since this inequality was not required along the sequence, or (b) there is a pair (J^{+}, J^{−}) such that θ_{j}^{n} → 1, ∀ j ∈ J^{+}, and

( ∑_{J^{+}} r_{sj} − ∑_{J^{−}} r_{sj}) > 0 and ( ∑_{J^{−}} r_{sj}δ_{sj}^{n} − ∑_{J^{−}} r_{sj})_{s} ≥ 0 along the sequence. In case (b),

by (iii) in lemma 4, we know that ∑_{J^{+}} q_{j}^{n} − ∑_{J^{−}} q_{j}^{n} > 0, but, at the cluster point we know only that the weak inequality holds, which, by (iv) in lemma 4, is not enough to prevent arbitrage. All the other cases, lead to the fullfilment of the desired weak inequalities at the cluster point.

Now, in case (a), (y_{n}^{h}) is bounded, for any consumer h, by the remark following lemma 3, and let y^{h} be its cluster point. In case (b), we can try to find a cluster point for (y_{n}^{h}) by inverting a Cramer subsystem of budget constraints (since θ_{j} = 1 for j ∈ J^{+}), provided that we already know that (x_{n}^{h}) has itself a cluster point.

Fatou’s lemma, part I, establishes the existence of an integrable function x mapping into ℝ_{+}^{L(S+1)} such that ∫_{I} x^{h} dλ ≤ lim_{n} ∫_{I} x_{n}^{h} dλ (passing to a subsequence if necessary) and x^{h} is a cluster point of (x_{n}^{h}) for a.e. h. Then, in case (b), we can pick # J^{+} + # J^{−} states and solve the respective Cramer system of budget constraints: p_{s} · (x_{s}^{h} − w_{s}^{h}) = ∑_{j∈J^{+}∪J^{−}} r_{sj}y_{j}^{h}− ∑_{j∉J^{+}∪J^{−}} G_{sj}^{h}, noticing that the last term is bounded, by absence of arbitrage opportunities involving these assets. The solution (y_{j}^{h})_{j∈J^{+}∪J^{−}} is a cluster point of the sequence (y_{j}^{hn})_{j∈J^{+}∪J^{−}}.

In fact, apply the implicit function theorem to the system of # J^{+} ∪ # J^{−} equations

f_{s}( ∑_{j∈J^{+}∪J^{−}} G_{sj}^{h} + ∑_{j∉J^{+}∪J^{−}} G_{sj}^{h}) − p_{s} · (x_{s} − w_{s}) = 0. We know that at the cluster point

(p, q, θ, t, η, c) this system has a solution at which the Jacobian with respect to y_{j}, j ∈ J^{+} ∪J^{−}, is nonsingular. Hence, there is a continuous function φ mapping a neighborhood

V of (p, q, θ, t, η, c, ∑_{j∉J^{+}∪J^{−}} G_{sj}^{h}, x) into a neighborhood U of (y_{j})_{j∈J^{+}∪J^{−}} and this function

describes the unique solution of the system in these neighborhoods. Now, for n sufficiently

large, (p^{n}, q^{n}, θ^{n}, t^{n}, η^{n}, c^{n}, ∑_{j∉J^{+}∪J^{−}} G_{sj}^{hn}, x^{n}) ∈ V and we must have (y_{j}^{hn})_{j∈J^{+}∪J^{−}} given

by φ. The continuity of φ implies that (y_{j}^{h})_{j∈J^{+}∪J^{−}} is a cluster point of (y_{j}^{hn})_{j∈J^{+}∪J^{−}}.

We want to show that (x^{h}, y^{h}) is an optimal solution of consumer h’s problem and that, therefore, the cluster point (q, θ, t, η) does not allow for arbitrage. Budget feasibility follows by closed graph of the budget correspondence and optimality uses the lower hemicontinuity of this correspondence (as in the proof of Theorem 1).

We are now ready to infer that the sequence of equilibria allocations of the truncated economies is uniformly bounded, which will allow us to apply Fatou’s lemma (part II, see appendix).

Lemma 6. If (p^{n}, q^{n}, θ^{n}, t^{n}, η^{n}, c^{n}) converges to (p, q, θ, t, η, c) and both the sequence (q^{n}, θ^{n}, t^{n}, η^{n}) and its limit point are contained in the set of nonarbitrage vectors, then the sequence of demanded bundles and portfolios (x^{hn}, y^{hn}) is uniformly bounded.

Proof: Let f_{0}(y; q, θ, t, η, c) = −q · y − t ∑_{s} η_{s}(f_{s} − R_{s}y). Suppose (y^{hn}) is not uniformly bounded. Then ess sup_{h}(y^{hn}) → ∞, for some j ∈ {1, …, A}. By lemma 3 we can decompose the nonuniformly bounded assets into a pair (J^{+}, J^{−}) of subsets of assets such that ∑_{J^{+}} r_{sj}δ_{sj} −∑_{J^{−}} r_{sj} ≥ 0, ∀ s. Now, ess sup_{h}f_{0}(y^{hn}) → −∞, either because condition (iii) in lemma 4 holds in the limit and ess sup_{h} t ∑_{J^{+}} ∑_{s:θ_{sj}<1} η_{s}G_{sj} → +∞ or because condition (iv) in lemma 4 holds in the limit. Then, −f_{0}(y^{hn}; q, θ, tη, c) > sup_{ℓ} ess sup_{h} w_{oℓ}^{h}, for h belonging to a set H of consumers of positive measure and n > N. Then, −f_{0}(y^{hn}; q^{n}, θ^{n}, t^{n}, η^{n}, c^{n}) > sup_{ℓ}ess sup_{h} w_{oℓ}^{h}, for h ∈ H and n sufficiently large, contradicting budget feasibility along the sequence.

Lemma 7. There exists an integrable function (x, y) mapping into ℝ_{+}^{L(S+1)} × ℝ^{A} such that ∫_{I} x^{h} dλ(h) = ∫_{I} w^{h} dλ(h), ∫_{I} y^{h} dλ(h) = 0, (x^{h}, y^{h}) is a cluster point of (x^{hn}, y^{hn}), for a.e. h, and (3) holds.

Proof: By lemma 6, we can apply Fatou’s lemma (part II, see appendix) to the se-

quence (z^{n}) ≡ (x^{n}, y^{n}, z̃^{n}) where z̃^{n} = ((y^{+})^{n}, τ^{n}, G^{n}, (G^{+})^{n}, f^{n}, (f_{s}^{n} − ∑_{j} G_{sj}^{n})τ_{sj}^{n} − (1 −

θ_{sj}^{n})r_{sj}(y_{j}^{+})^{n}, (G_{sj}^{+})^{n} − θ_{sj}^{n}r_{sj}(y_{j}^{+})^{n}). Then, there is an integrable function z such that ∫_{I} z(h)dλ(h) = lim_{n} ∫ z^{n}(h)dλ(h) and, for almost every h, z(h) is a cluster point of (z^{n}(h)). Now, z(h) = (x(h), y(h), z̃(h)) and, by continuity of the functions involved, z̃(h) must be equal to (y^{+}, τ, G, G^{+}, f, (f_{s} − ΣG_{sj})τ_{sj} − (1 − θ_{sj})r_{sj}y_{j}^{+}, G_{sj}^{+} − θ_{sj}r_{sj}y_{j}^{+}) evaluated at y(h). This implies that markets clear and equation (3) holds, for any (s, a). Moreover (x(h), y(h)) is an optimal solution, as in the end of the proof of lemma 6.

This completes the proof of the theorem. We close this section with two remarks on the role of some assumptions.

Remark 2. Without assumption (A4), the default barrier established in lemma 3 would not exist and one of the nonarbitrage conditions would be as follows. For any pair (J^{+}, J^{−}) of subsets of assets, such that (i) ∑_{j∈J^{+}} r_{sj}δ_{sj} − ∑_{j∈J^{−}} r_{sj} ≡ r_{s} is negative for s ∈ S_{1} ≠ ∅, positive for s ∈ S_{3} ≠ ∅ and possibly zero for s ∈ S_{2} , but (ii) θ_{sj} = 1, ∀ j ∈ J^{+},

s ∈ S_{2} ∪ S_{3} , we must have ∑_{j∈J^{+}} q_{j} − ∑_{j∈J^{−}} q_{j} > t( ∑_{s∈S_{2}∪S_{2}} η_{s} ∑_{j∈J^{+}} r_{sj} − ∑_{s} η_{s} ∑_{j∈J^{−}} r_{sj}). This

strict inequality condition would create a problem when we would try to redo lemma 5, since the condition accommodates a case where bought assets do not reimburse completely (i.e., θ_{sj} < 1), for some s ∈ S_{1} . In fact at a cluster point, only the weak inequality is guaranteeed to hold, but we might not find a Cramer subsystem of budget constraints to solve for (y_{J^{+}}, y_{J^{−}}).

Remark 3. If the returns function G_{sj} were piecewise linear (that is, if G_{sj} were given by θ_{sj}r_{sj}y_{j}, for y_{j} > 0, as in the proportional reimbursement scheme adopted by Dubey-Geanakoplos-Shubik and Araujo-Monteiro-Pascoa), the nonarbitrage conditions (under assumption (A4) and a default barrier established as in lemma 3) would be as follows. For any pair (J^{+}, J^{−}) of subsets of assets, such that ∑_{j∈J^{+}} r_{sj}θ_{sj} − ∑_{j∈J^{−}} r_{sj} ≡ r_{s} ≥ 0, for any

s, and r_{s} > 0, for some s, we must have ∑_{j∈J^{+}} (q_{j} − t ∑_{s:θ_{sj}<1} η_{s}(1 − θ_{sj})) > ∑_{j∈J^{−}} q_{j}. This

strict inequality would create a problem when we would try to redo lemma 5. In fact, at a cluster point only the weak inequality is guaranteed to hold, but we might not find a Cramer subsystem of budget constraints to solve for (y_{J^{+}}, y_{J^{−}}), if θ_{sj} were less than one for some j ∈ J^{+}. This is a case of ex-post redundancy of assets, due to default.

3.7 More General Repayment Rules

Theorem 2 above on existence of equilibria still holds if we replace the specific form of the nonproportional reimbursement rule G_{sa} by a general form satisfying certain axioms. In this section we will address this extension and appeal to the extensive literature on bankruptcy found in the cooperative game theory. In this literature, the bankruptcy problem can be formally described as a pair (x, T) where x denotes the vector of n claims on an estate having total value T ≤ ∑_{i} x_{i}. A solution is a vector t ∈ R^{n} such that

∑_{i} t_{i} = T and 0 ≤ t_{i} ≤ x_{i} for all i.

The class of solution methods that has received more attention is the parametric class (see Moulin (1987) and Young (1988)). Here the solution t_{i} assigned to each agent is given by a function f that depends on the pair (x, T) only through the individual claim x_{i} of the agent and a parameter μ chosen so that ∑_{i} t_{i} = T. Clearly, this parametric methods are compatible with a general equilibrium frame work, where each agent should not be concerned with other agents’ claims and should take the parameter μ as given. Young (1987) showed that a continuous solution method F: (x, T) ↦ t is representable by a continuous parametric function f if and only if it is anonymous and pairwise consistent.

Anonymity requires that allocations should be determined by individuals’ claims, not by who they are (that is, x_{i} = x_{j} ⇒ t_{i} = t_{j}); pairwise consistency requires that, for any pair J of agents, if t = F (x, T), then t_{J} = F (x_{J}, ∑_{i∈J} t_{i}). In the presence of continuity and anonymity, pairwise consistency actually implies consistency: the above condition holds for any subset J, not just for pairs of agents. Consistency is clearly a desirable property stating that the allocation should be invariant to restricting the problem to a subset of agents.

Examples of parametric methods include the proportional method (given by t_{i} = μx_{i}), the constained equal award (CEA) method (given by t_{i} = min{μ, x_{i}}) and the constrained equal loss (CEL) method (given by t_{i} = max{0, x_{i} − μ}). The CEA rule makes awards as equal as possible, subject to the condition that no agent receives more than his claim, whereas the CEL rule makes losses as close as possible, subject to the condition that no creditor ends up with a negative award (see Herrero-Villar (1998) on the properties of these two rules).

We will require the reimbursement rule G_{sa} to be parametric, continuous of the form G_{sa}(r_{sa}y_{a}, μ_{sa}) where μ_{sa} is such that equation (3) holds, which is the equation requiring the garnished estates to be equal to the reimbursements to creditors. Furthermore, we will assume regressiveness of retunrs: the marginal rate of reimbursement should tend to zero as the claim goes to infinity. To simplify, let us assume that G_{sa} is differentiable in the rest argument, at least beyond a certain level, and require:

lim_{y_{a}→∞} ∂G_{sa}(r_{sa}y_{a}, μ_{sa})/∂y_{a} = 0.

Clearly, CEA and the particular form considered in section 3.1 satisfy this requirement (in the case of CEA, the partial derivative is identically zero for y_{a} > μ_{sa}/r_{sa}).

Actually, for equation (3) to be satisfied it surfaces that μ_{sa} be such that

∫ (G_{sa}^{h})^{+} dλ(h) ≥ θ_{sa}r_{sa} ∫ (y_{a}^{h})^{+} dλ(h)

where θ_{sa} is the mean reimbursement rate (see equation (3’)). As shown in the proof of Theorem 2 (see section 3.3), the above inequality implies equation (3), due to the specification taken for the spread.

The proof of Theorem 2 is still valid. Notice that even thought G_{sa} might not be concave in μ_{sa} (as was the case for the specific function introduce in section 3.1), the above inequality can still be satisfied by choosing μ_{sa} appropriately.

The CEL rule is not regressive and neither is the solution proposed by Aumann-Maschler (1985). These authors recalled the Talmudic solution to the problem of sharing as estate among the three wives of a deceased and showed that this function is v(J) = max {0, T − ∑_{i∉J} x_{i}}, for any subset J of agents. The solution contemplated by Aumann and Maschler is consistent but may be a concave or a convex function of the individual claim, depending on the relative magnitude of the estate and the claims. It remains unknown whether the solution proposed by Aumann and Maschler is compatible with the existence of equilibria without bounded short-sales.

3.8 An Example

As explained above, nonproportional returns determine that positive financial returns may be strictly concave whereas negative returns are always linear and, for this reason, tend to dominate as a first-period budget-feasible portfolio increases unboundedly. Thus, an illustrative example should have at least two assets. Suppose S = 3 and A = 2 and the nominal returns are (1,1) in the first state, (1,0) in the second state and (0,1) in the third state. There is a single physical good and we assume that there is no consumption and no endowment in the first period.

The economy has two types of consumers, with the same weight in the economy, whose preferences are the same, given by u = ℓn x_{1} + ℓn x_{2} + ℓn x_{3} , and whose endowments are (φ_{1}^{a}.φ_{2}^{a}, φ_{3}^{a}) = (10, 5, 15) for consumer a and (φ_{1}^{b}.φ_{2}^{b}, φ_{3}^{b}) = (10, 15, 5) for consumer b. We will construct an equilibrium where consumer a buys asset 1 and sells asset 2, transfering income from state 1 to state 2 and from state 3 to state 1. Consumer b will buy asset 2 and sell asset 1, transfering income from state 1 to state 3 and from state 2 to state 1. In this equilibrium consumer a goes bankrupt in state 3 and consumer b goes bankrupt in state 2. The spot price is set equal to one in every state.

The liability rule is given by γ(z) = 1/3 for z ∈ [−\frac{2}{3} ω_{s}, −\frac{1}{3} ω_{s}] and γ(z) = γ̃ + (1 − γ̃)(1 − exp\{−(z + \frac{2}{3} ω_{s}^{2}\}) for z < −\frac{2}{3} ω_{s} where γ̃ is slightly over 2/3 say equal to 0.66. When z > −\frac{1}{3} ω_{s} we have f_{s} = ∑_{j} G_{sj}. Assumption (A4) of Theorem 2 is satisfied. Figure 5 illustrates this liability rule for ω_{s} = 15.

Fig. 5

The nonproportional reimbursement rule is the constrained equal award rule presented in secion 3.7. A creditor with a claim c_{i} receives min{c_{i}, λ} where λ is such that ∑_{i} min{c_{i}, λ} equals the estate available for partition.

The first period budget constraint is πy + ∑_{s} β_{s}(f_{s} − R_{s}y) = 0. Due to the symmetry of asset returns and endowments we will look for asset prices such that π_{1} = π_{2.}. Let us set y_{1}^{a} = 10, y_{2} = −10, y_{1}^{b} = −10 and y_{2}^{a} = 10. Then, for consumer a we have 0 + β_{2}(5 − 10) + β_{3}(−5 − (−10)) = 0, that is, −5β_{2} + 5β_{3} = 0, and for consumer b we have 0 + β_{2}(−5 + 10) + β_{3}(5 − 10) = 0, that is, 5β_{2} − 5β_{3} = 0. Then, β_{2} = β_{3} . Both π and β are indeterminate. Let us assume that π ≥ β. The corresponding real allocation is x^{a} = (10, 5 + λ, 15 − 5) and x^{b} = (10, 15 − 5, 5 + λ) where λ = 5.

It remains to check that the portfolio choice is individually optimal. No consumer has an incentive to sell more than 10 units of the asset that he is selling since income would fall from 10 to a value slightly above 5 in the bankruptcy state and at the same time the extra purchase of the other asset can not increase his income over ω_{s} + λ in the state where he suffers from bankruptcy given by others. Moreover, this local optimum is actually global. On the one hand, the consumer wishes to flaten the consumption pattern over states and would not like to transfer lower amounts across states. In fact, (i) utility remains unchanged if the absolute value of the assets transactions drop to levels between 5 and 10, (ii) for π ≥ β, by the first period constraint, if the short-sale falls below 5 then the purchase of the other asset should also fall below 5, but, in this case, u = ℓn 10 + ℓn(5 + t) + ℓn(15 − t), where t is the common value of the short-sale and the purchase, and u′_{t} is negative for t < 5. On the other hand, higher transfers lead to seizures of the estate in the bankruptcy state that are too high (comparing with the gains in the other state, since for z < −\frac{2}{3} ω_{s} we have u = 2ℓn 10 + ℓn 15 + ℓn(1 − γ̃) − (z + \frac{2}{3} ω_{s}^{2} which decreases as the financial return z in the bankruptcy state falls, that is, as y_{2}^{a} falls below −10 or y_{1}^{b} falls below −10).

4. CONSTRAINED EFFICIENCY

We will discuss now the efficiency properties of equilibria. We will show in Proposition 1 that an equilibrium allocation is always weakly constrained efficient and in Proposition 2 that, generically, it will not be strongly constrained efficient. Since proofs are a bit harder for the nonproportional reimbrusement model, we deal only with this case in the proofs.

Proposition 1. Let (x̄, ȳ, p̄, q̄, θ̄) be an equilibrium (together with a vector (t̄, η̄, c̄) in de case of section 3 above). The allocation (x̄, ȳ) is efficient among all allocations (x, y) such that

(i) ∫_{I} x^{h} dλ(h) = ∫_{I} ω^{h} dλ(h) and ∫_{I} y^{h} dλ(h)

(ii) p̄_{s} · (x_{s}^{h} − ω_{s}^{h}) = f̄_{s}^{h}(y^{h}) ≡ max ( ∑_{j=1}^{A} G_{sj}(r_{sj}y_{j}), − ∑_{ℓ=1}^{L} p̄_{sℓ}γ_{sℓ}ω_{sℓ}^{h}), for any s = 1, …, S

and a.e. h

(iii) p̄_{0} · (x_{0}^{h} − ω_{0}^{h}) + q̄ · y^{h} + T^{h} = 0, for a.e. h, where

1. (iv) T ∈ L^{1}(I, ℝ) is such that ∫_{I} T^{h} dλ(h) = 0.

Proof: Suppose there exists an allocation (x, y) which, together with a transfers function T, satisfies conditions (i) through (iv) and u^{h}(x^{h}) > u^{h}(x̄^{h}) for a.e. a. Then the first period budget constraint must be violated, that is,

p̄_{0} · (x_{0}^{h} − ω_{0}^{h}) + q̄ · y^{h} + t̄∑_{s} η̄_{s}(f̄_{s}^{h} − R_{s}y^{h}) > 0, h ∈ I.

Now, aggregating,

p̄_{0} · ∫_{I}(x_{0}^{h} − ω_{0}^{h})dλ(h) + q̄ · ∫_{I} y^{h} dλ(h) + t̄∑_{s} η̄_{s} ∫ (f̄_{s}^{h} − R_{s}y^{h})dλ(h) > 0.

But we know that ∫_{I}(x_{0}^{h} − ω_{0}^{h})dλ(h) = 0, ∫_{I} y^{h} dλ(h) = 0 and R_{s} ∫_{I} y^{h} dλ(h) = 0. Then, ∑_{s} η_{s} ∫_{I} f̄_{s}^{h} dλ(h) > 0 and, therefore, there is a state s for which ∫_{I} f̄_{s}^{h} dλ(h) > 0. Now, by (ii) we must have ∫_{I}(x_{sℓ}^{h} − ω_{sℓ}^{h})dλ(h) > 0 for some ℓ = 1, …, L, contradicting (i).

Remark 4. It is interesting to notice the role played by the spread in Proposition 1. The weakly constrained efficient transfer among agents is given by the equilibrium spread t̄∑_{s} η̄_{s}(f̄_{s}^{h} − R_{s}ȳ^{h}), which is a discounted expected value of the difference between the consumer’s default and the default suffered by the consumer.

Our default model can be seen as a model of externalities, where default by others decreases returns according to the function G_{sj}. The inefficiency introduced by this externality can be corrected when a market for the externality is created, assuming that legal rights stipulate that agents are entitled to full reimbursement of claims. In our model, there is a market for the externality in each state and the respective price is given by tη_{s}. As suggested by Coase (1960), the resulting market equilibrium is free from the inefficiency caused by the externality.

There is, however, another source of inefficiency, which is the incompleteness of markets. We will examine now the performance of the equilibrium allocations in the light of a stronger efficiency criterion.

The above weak constrained efficiency property of equilibria is analogous to the properties found in the incomplete markets model without default and also in the collateral model (without utility penalties) of Geanakoplos-Zame-Dubey. As in these two models, the equilibria of our model will, generically, be dominated when prices are no longer assumed to be kept constant at the equilibrium levels.

Proposition 2. Generically (in endowments and utility functions), equilibrium allocations are inefficient among the allocations (x, y) for which there exist a transfer function T ∈ L^{1}(I, ℝ) and prices (p, q) ∈ ℝ_{+}^{L(S+1)} × ℝ_{+}^{A} satisfying

(i) ∫_{I} x^{h} dλ(h) = ∫_{I} ω^{h} dλ(h) and ∫_{I} y^{h} dλ(h) = 0

(ii’) p_{s} · (x_{s}^{h} − ω_{s}^{h}) = f_{s}^{h}(y^{h}) ≡ max ( ∑_{j} G_{sj}(r_{sj}y_{j}), − ∑_{ℓ} p_{sℓ}γ_{sℓ}ω_{sℓ}^{h}), for any s = 1, …, S

and a.e. h

(iii’) p_{0} · (x_{0}^{h} − ω_{0}^{h}) + q · y^{h} + T^{h} = 0, for a.e. h, (iv’) ∫_{I} T^{h} dλ(h) = 0.

Proof: It is enough to show that an equilibrium allocation does not satisfy the necessary first order condtions of the following welfare maximization problem, given a ∈ L^{1}(I, ℝ_{+}),

max_{(x,y,T,p,q)} ∫_{I} a^{h}u^{h}(x^{h})dλ(h)

subject to (i), (ii’), (iii’) and (iv’).

Let ρ_{s}^{h} be the Lagrange multiplier of constraint (ii’)_{h,s} and let ρ_{0}^{h} be the Lagrange multiplier of constraint (iii’)_{h}. Then, setting the derivative of the Lagrangean with respect to p_{sℓ} equal to zero we obtain the following equation:

(4)  ∫_{I} ρ_{s}^{h}[(x_{sℓ}^{h} − ω_{sℓ}^{h}) + γ_{sℓ}ω_{sℓ}^{h}χ_{B_{s}}^{h}]dλ(h) = 0

where B_{s} = {h: f_{s}^{h} = − ∑_{ℓ} p_{sℓ}γ_{sℓ}ω_{sℓ}^{h}}.

Let us check if this necessary condition for efficiency can be satisfied in equilibrium. Notice that the first order conditions with respect to (x, y) are satisfied at an equilibrium (x̄, ȳ, p̄, q̄, θ̄, t̄, η̄, c̄) (answer already know, by Proposition 1), when we set the multipliers of the commodity-feasibility constraints equal to t̄η̄_{s}p̄_{s} and ρ_{s}^{h} = \frac{λ_{s}^{h}}{λ_{0}^{h}} − t̄η̄_{s} (s = 1, …, S), where λ_{s}^{h} = ||∇u_{s}^{h}(x_{s}^{h})||/||p̄_{s}|| (s = 0, 1, …, S). Replacing in equation (4) we obtain:

∫_{I} \frac{λ_{s}^{h}}{λ_{0}^{h}} [(x̄_{sℓ}^{h} − ω_{sℓ}^{h}) + γ_{sℓ}ω_{sℓ}^{h}χ_{B_{s}}^{h}]dλ(s) =

t̄η̄_{s} ∫_{I}(x̄_{sℓ}^{h} − ω_{sℓ}^{h})dλ(h) + t̄η̄_{s}γ_{sℓ} ∫_{B_{s}} ω_{sℓ}^{h} dλ(h) (5)

Notice that the first integral on the right hand side is zero and that when B_{s} is of measure zero the equation becames ∫_{I} \frac{λ_{s}^{h}}{λ_{0}^{h}} (x̄_{sℓ}^{h} − ω_{sℓ}^{h})dλ(h), which was the necessary condition for efficiency in the incomplete markets model without default (see Geanakoplos-Polemarchakis (1986) and Magill-Shaffer (1990)). Just like this condtion was generically (in endowments and utility functions) violated in equilibria of the model without default, now equation (5) will not hold, generically, in equilibria. (since the LS equations (5), together with the equilibrium equations, constitute an overdetermined system).

APPENDIX

Fatou’s lemma (in m m m-dimension)

I) Let (f_{n}) be a sequence of integrable functions of a measure space (Ω, A, ν) into ℝ_{+}^{m}. Suppose that lim_{n} ∫_{Ω} f_{n} dν exists. Then there is an integrable function f of Ω into ℝ_{+}^{m} such that f(w) is a cluster point of (f_{n}(w)), for a.e. w, and ∫_{Ω} f dν ≤ lim_{n} ∫_{Ω} f_{n} dν.

II) Let (f_{n}) be a uniformly integrable sequence of integrable functions of a measure (Ω, A, ν) into ℝ^{m}. Suppose also that lim_{n} ∫ f_{n} dν exists. Then there is an integrable function f of Ω into ℝ_{+}^{m} such that f(w) is a cluster point of (f_{n}(w)), for a.e. w, and ∫_{Ω} f dΩ = lim_{n} ∫_{Ω} f_{n} dν (see Arstein (1979) or Hildenbrand-Mertens (1971)).

Lemma: Let G: ℝ_{+} → ℝ given by G(z) = z(1 − exp(−δz^{−γ})), where δ = c^{γ}(1 − γ)/γ, γ ∈ (0, 1) and c > 0. Then dG/dc > 0 and d^{2}G/dc^{2} < 0.

Proof:

\frac{dG}{dc} = z exp(−δz^{−γ})z^{−γ} γ c^{γ−1}(1 − γ)/γ > 0

\frac{d^{2}G}{dc^{2}} = z[z^{−γ}(\frac{dδ}{dc})^{2} exp(−δz^{−γ})(−z^{−γ}) + exp(−δz^{−γ})z^{−γ} \frac{d^{2}G}{dc^{2}}]

now \frac{d^{2}δ}{dc^{2}} = γ(γ − 1)c^{γ−2}(1 − γ)/γ < 0 and therefore

\frac{d^{2}G}{dc^{2}} = z[ − z^{−2γ}(\frac{dδ}{dc})^{2} exp(−δz^{−γ}) + exp(−δz^{−γ})z^{−γ} \frac{d^{2}δ}{dc^{2}} ]

is negative.

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