Economia Financiera Avanzada

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1 Model Economia Financiera Avanzada EBAPE- Fundação Getulio Vargas Universidad del Pacífico, Julio 5 21, 2011 Economia Financiera Avanzada

2 Model Default and Bankruptcy in GE Models Economia Financiera Avanzada

3 Model Introduction The possibility of default creates a need for the introduction of incentives (enforcement mechanisms) for agents to keep their promises. It is certainly not possible in practice to devise a strong enough mechanism that ensures that all promises will be kept in all circumstances. It is often the case that this is also not desirable economically. In fact, the main effect of such may be indistinguishable a simple restrictions on trade which prevent efficient risk sharing. Economia Financiera Avanzada

4 Model Default and Bankruptcy in GE Models Uncertain deliveries on contracts. Penalties: Shubik, M. and Wilson, C. (1977). The optimal bankruptcy rule in a trading economy using fiat money, Z. Nationalokon. 37, Utility penalties: Dubey, Geanakoplos and Shubik (2005), Default and Punishment in General. Econometrica, 73(1), Efficiency: Zame, W.R.,(1993), Efficiency and the role of default when security markets are incomplete, American Economic Review 6, Collateral seizure: Dubey, Geanakoplos and Zame (1995). Default, collateral and derivatives, Yale University, mimeo. Geanakoplos and Zame (2010). Collateralized Asset Markets. Economia Financiera Avanzada

5 Model Fundamental Theorem of Asset Pricing with Default and Collateral Fajardo (2005) and Orrillo (2005): Exogenous Collateral. Araujo, Fajardo and Pascoa (2005): Endogenous Collateral. Economia Financiera Avanzada

6 Model Model Utility Penalties and Collateral Agents are allowed to default and suffer utility penalties Agents need to put collateral when they short sales No Tranching nor Pyramiding allowed Anonymous Market: Penalties and Collateral Economia Financiera Avanzada

7 Model Model Utility Penalties and Collateral Exchange economy over two periods. Finite number of states s S = {1, 2,..., S}. H agents J assets L durable goods. In the first period, there is a market where physical commodities and assets are traded against each other. In the second period asset returns are delivered. Economia Financiera Avanzada

8 Model Model Let θ j 0 be the number of units of asset j the consumer bought. ϕ j 0 be the number of units he sold.. Every sale should be backed by a bundle of goods (collateral): C j IR L + \ {0}. Asset j is defined by the promise of goods R j IR L + \ {0} it makes and the collateral backing it: (R j, C j ) The collateral in this model is keep by the borrowers, in this way they will have utility returns from the use of collateral, this is the case of the Collateralized Mortgage Obligation markets (CMO) Economia Financiera Avanzada

9 Model Model ω h = (ω0 h, (ωh ) s S ) IR + L IR + SL, initial endowments of agent h, such that es h 0, s S {0}. p = (p 0, (p s ) s S ) L 1 SL 1 commodity price system and π J 1 asset prices. (R j, C j ) j J are real assets. Y : S IR L ++ depreciation rates (durability of goods):y (s)c j is the depreciated bundle of goods. Each C jl in the collateral bundle C j will produce a depreciated good Y l (s)c jl in the second period. Economia Financiera Avanzada

10 Model Model U h : IR + L IR + SL IR is the utility function of agent h. x = (x 0, (x s ) s S ) is the consumption plan, x 0 and (x 1, x 2,.., x S ) first and second period consumption. λ h sj IR + utility penalty for each dollar of default by the borrower h on asset j in state s. dj h : S IR L + true return on asset j in state s. i.e. d h j (s) bundle to be delivered by agent h on asset j in state s. (Depends on λ) K j (s) expected bundle to be delivered by each unit of asset j in state s. Economia Financiera Avanzada

11 Model Model Our economy is defined by E = ((U h, ω h ) h H, (R j, C j, (λ h j ) h H) j J, (Y l ) l L ) Budget set is given by: p 0 (x 0 ω0) h + π(θ ϕ) + p 0 C j ϕ j 0, (1) p s (x s ωs h Y (s)x 0 ) ϕ j p s [Y (s)c j ] p s K j (s)θ j + p s d j (s) 0, s j J j J j J (2) min{r jl (s), Y (s)c jl } d jl (s), s S, j J, l L. (3) j J Economia Financiera Avanzada

12 Model Individual Optimization Problem Each agent h H face the following problem: max U h (x 0 + Cϕ, x 0 ) (x,θ,ϕ,d) B h (p,π,k ) s j λ h sj ( ps R j (s)ϕ j p s d j (s) ) + p s v s (4) where B h : L 1 SL 1 J 1 SJ 1 IR L(S+1) + IR J + IR J + IR SJL + is defined by: B h (p, π, K ) := {(x, θ, ϕ, d) IR L(S+1) + IR J + IR J + IR SJL + : (1), (2) and (3) hold}, this budget set is convex and v s IR L + \ {0} is an exogenous market bundle used to measure disutility in real terms. Economia Financiera Avanzada

13 Model Matrix form Now lets use the following matrix form: P ( x ω h ) [ Π A ] Ψ, where P ( x ω h ) = ( p 0 ( x 0 ω 0 ),..., p S ( x S ω S ) ), x = (x 0, x 1 + j d j(1),..., x S + j d j(s)), ω = (ω 0, ω 1 + Y 1 x 0,..., ω S + Y S x 0 ),Ψ = (θ, ϕ), Π = (π, p 0 C π) and p 1 K (1) p 1 Y (1)C p 2 K (2) p 2 Y (2)C A (D) = p S K (S) p S Y (S)C Economia Financiera Avanzada

14 Model Definition An equilibrium for E = ((U h, ω h ) h H, (R j, C j, (λ h j ) h H ) j J, (Y l ) l L ) is a price vector (p o, p, π) and an allocation (x h, θ h, ϕ h, d h ) h H such that: 1 Allocations (x h, θ h, ϕ h, d h ) maximize utility functions subject to budget set B(p o, p, π). 2 Markets clear: (x h o + Cϕh ω h o ) = 0, H (x h s ωh s Ysxh o YsCϕh ) = 0, s S, H (θ h ϕ h ) = 0, H 3 H K h j (s)θ h j = H d h j (s)ϕ h j in each state s and asset j. Economia Financiera Avanzada

15 Model FTAP with Default and C = 0 Not possible to define arbitrage in micro terms. If agents maximize utilities, arbitrage free prices must exist Economia Financiera Avanzada

16 Model and Efficiency with Default and C = 0 By imposing bounds on arbitrage free prices DGS (2005), prove existence of equilibrium. Moreover, they show numerically that it is possible to create a Pareto improvement by allowing agents to default. Drawback λ not observable. Economia Financiera Avanzada

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18 Model No Utility Penalties λ = 0 In this model D j (s) := min{p(s)r j (s), p(s)y (s)c j } is the true return on asset j in state s. Our economy is defined by E = ((U h, ω h ) h H, (R j, C j ) j J, (Y l ) l L ) the budget constraints of each agent are: p 0 (x ω h 0 ) + π(θ ϕ) + p j J C j ϕ j 0, (5) p(s)(x(s) ω h (s) Y (s)x) ϕ j p(s)[y (s)c j ] j J j J (θ j ϕ j )D j (s) 0, (6) Economia Financiera Avanzada

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20 Model Individual Problem In this setting each agent h H face the following problem: where max U h (x + Cϕ, x 0 ) (7) (x,x 0,θ,ϕ) B h (p,π) B h : L 1 L 1 J 1 IR L + IR SL + IR J + IR J + is defined by: B h (p, π) := {(x, θ, ϕ) IR L + IR SL + IR J + IR J + : (5) and (6) hold}, Economia Financiera Avanzada

21 Model Budget Set [ P (x ω h Π ) A (C) ] Ψ, P (x ω) = (p 0 (x 0 ω 0 ), p(1)(x 1 ω 1 Y 1 x 0 ),.., p(s)(x S ω s Y S x 0 )), Ψ = (θ, ϕ),π = (π, pc π) and D(1) p(1)y 1 C D(1) D(2) p(2)y 2 C D(2) A (C) = D(S) p(s)y S C D(S) D(s) = (D 1 (s), D 2 (s),.., D J (s)) is the vector of true returns. π and π p 0 C are the vectors of theeconomia buy price Financieraand Avanzada the net sale

22 Arbitrage Model Let us start by defining arbitrage opportunities in a nontrivial context where p 0 >> 0 and p(s) >> 0, s. Definition We say that there exists strong arbitrage opportunities if Ψ IR 2JL + such that ΠΨ < 0 and A(C)Ψ 0 Definition We say that Ψ IR + 2JL is an arbitrage opportunity if it is either a strong arbitrage or is such ΠΨ = 0 and AΨ > 0. i.e. ΠΨ 0 and A (C) Ψ 0 with at least one strictly inequality. Economia Financiera Avanzada

23 Model FTAP with Exogenous Collateral Theorem a) There is no strong arbitrage if and only if there exist β IR + S such the inequalities in (8) are satisfied. b) There is no arbitrage if and only if there exist β IR ++ S such the inequalities in (8) are satisfied. S β s D j (s) π j (p 0 s=1 S β s p(s)y s )C j + s=1 S β s D j (s) (8) Inequality (8) tell us that arbitrage free prices have a spread that takes in account the cost of collateral depreciation and the fact that promises can be not fully delivered, since D j (s) = p(s)r j (s) (p(s)r j (s) p(s)y (s)c j ) +. s=1 Economia Financiera Avanzada

24 Proof Model Construct the following matrix: D(1) p(1)y 1 C D(1) D(2) p(2)y 2 C D(2) Â = D(S) p(s)y S C D(S) I 0 0 I Where I is the J J identity matrix and 0 is the J J null matrix. We can observe that y : Ây 0 y : Ay 0 and y 0. Economia Financiera Avanzada

25 Proof Model Then absence of strong arbitrage is equivalent to / y IR 2J such that Ây 0 and Πy < 0. Now by the Farkas Lemma it is equivalent to β = (β 1,.., β S+2J ) IR + S+2J such that: from here we obtain: Â β = Π, j S = β s D j (s)+β S+j and p 0 C j π j = S S β s p(s)y s C j β s D j (s)+β S+J+ s=1 s=1 s=1 since β is positive we obtain the desired inequalities. (9) In analogous way we obtain (b) using another version of Farkas Lemma (see Luenberger (1969), pag. 167). Economia Financiera Avanzada

26 Remarks Model Another consequence of the above Theorem is p 0 C j π j 0 j J with strict inequality if C j 0. Observe that pc j = π j creates also arbitrage opportunities since even if p(s)y s C = D j (s) for every s there would be unbounded utility gains from consumption of C j ϕ j by choosing unbounded short sales of asset j. Economia Financiera Avanzada

27 Model Second Part of FTAP Theorem Under the assumptions assumed on agents s utility functions, UMP has a solution only if there are no whether arbitrage or strong arbitrage on the financial markets. Conversely, if there is no arbitrage, then UMP has a solution. Economia Financiera Avanzada

28 Proof Model (i) (ii) Let the vector (x h, θ h, ϕ h ) be a solution to the optimization problem, then p o(x h o ωh o ) + πθh + (p oc π)ϕ h = 0 and p s(x h s ωh s Ysx h o ) Dsθh + (p sy sc D s)ϕ h, s S Now, suppose π allows for strong arbitrage opportunities. Then there is an arbitrage portfolio (θ, ϕ) R 2J + then, a new portfolio ( θ = θ h + θ, ϕ = ϕ h + ϕ) such that p o(x o ω h o ) + π θ + (π p oc) ϕ 0 and As a result, p s(x s ω h s Y sx o) D s θ + (psy sc D s) ϕ, s S. u h (x h o + C ϕ, x h o ) > uh (x h o + Cϕh, x h o ) since u h is strictly increasing with regard to first-period variables, and therefore a contradiction of the individual optimality of (x h, θ h, ϕ h ). For the case in which π allows no arbitrage opportunity the argument is the same. Economia Financiera Avanzada

29 Proof Model (ii) (i) It is sufficient to prove that the budget set at arbitrage-free prices is compact. As this is already closed, it remains to be shown that it is bounded. First, commodity prices are strictly positive, since U h is strictly increasing. This implies that the set of feasible consumption plans x R L(S+1) + is bounded. In fact, from (4), it follows that p ox o + s S β sd sθ + s S β s[p sy sc D s]ϕ p oω h o and p ox o + s S β sp sx s s S β sp s(ω h s + Y sx o) We now suppose that there is a feasible sequence (θ n, ϕ n ) such that (θ n, ϕ n ) 1. Thus, the sequence (θ n,ϕ n ) (θn, ϕ n ) is bounded and therefore admits a convergent subsequence whose limit, say (θ, ϕ), Budget feasibility implies that there exists a bounded sequence x n R L(S+1) + such that p o(x n o ωh o ) + πθn + (p oc π)ϕ n 0 (5) p s(x n s ω h s Y sx n o ) D sθ n + (p sy sc D s)ϕ n, s S (6) Dividing both sides of the previous equalities by (θ n, ϕ h ), and taking the limit as n, we obtain Economia Financiera Avanzada

30 Proof Model πθ + (p oc π)ϕ 0 D sθ + (p sy sc D s)ϕ 0, s S with (θ, ϕ) belonging to the unitary sphere of R+ 2J. Consider vectors β RS ++, of Item b of Theorem 1. Multiplying the last equality by β s then summing over s and finally using (4 ) we have πθ + (p oc π)ϕ = 0 Since both terms in the last equality are positive, each one is positive separately as well. That is, πθ = 0, (p oc π)ϕ = 0 From remarks of theorem 1 follows that both π j > 0, j and p oc j π j > 0, j. Thus (θ, ϕ) = (0, 0) contradicting the fact of belonging to the unitary sphere of R 2J. Thus, the budget set is compact and therefore the consumer s problem has a solution, since the utility function is continuous. Economia Financiera Avanzada

31 Model Existence Theorem Under the usual assumptions that utility functions are continuous, strongly monotonic and endowments are different from zero. The absence of arbitrage implies the existence of equilibrium, independent of the way that agents deliver. Economia Financiera Avanzada

32 Model Existence Theorem Under the usual assumptions that utility functions are continuous, strongly monotonic and endowments are different from zero. The absence of arbitrage implies the existence of equilibrium, independent of the way that agents deliver. Economia Financiera Avanzada

33 Proof Model Basically, absence of arbitrage allow us to obtain endogenous bounds on short sales. As we can see from budget set: p 0 x 0 + πθ + j J (p 0 C j π j )ϕ j p 0 e h 0, From FTAP for each j J, p 0 C j π j > 0. Then, ϕ j p 0e h 0 p 0 C j π j. As pointed out by Radner (1972), the failure of existence of equilibrium is due to discontinuities in the budget set, that can be avoided by putting bounds on short sales.. Economia Financiera Avanzada

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37 Model Efficiency with Collateral A. Araujo, F. Kubler and S. Schommer, 2011, Regulating Collateral Requirements when Markets are Incomplete. forthcoming JET. Kilenthong, W. T Collateral Premia and Risk Sharing under Limited Commitment. Economic Theory, 46, Kilenthong, W. T. and R. M. Townsend (2011). Market Based, Segregated Exchanges with Default Risk. WP Fajardo, J. (2011). Constrained Efficiency with Endogenous Collateral. WP Economia Financiera Avanzada

38 Introduction The model Numerical examples Conclusion General equilibrium with collateral Scarcity and the efficiency of risk-sharing Welfare effects of regulation Scarcity and the efficiency of risk-sharing A key feature of the model is that scarcity and an unequal distribution of collateralizable durable goods affects risk-sharing and welfare. If the durable good is plentiful, the model is equivalent to a standard Arrow-Debreu model (and competitive equilibrium allocations are Pareto-optimal). If, on the other hand, the collateralizable durable good is scarce, most assets are not traded in equilibrium and markets appear to be incomplete. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

39 Introduction The model Numerical examples Conclusion Can regulation improve welfare? General equilibrium with collateral Scarcity and the efficiency of risk-sharing Welfare effects of regulation In the presence of scarcity, a most interesting question is whether welfare improvements might be achieved through government regulation. It is a quantitative question who in the economy gains and who loses through a regulation of collateral-requirements. We provide a series of examples, some of them illustrative and some realistically calibrated, in order to address this question. The numerical examples illustrate that regulation of margin requirements generally does not lead to Pareto-improvements. However often a majority of agents would favor a regulation since it is welfare improving for them. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

40 Subprime regulation Introduction The model Numerical examples Conclusion General equilibrium with collateral Scarcity and the efficiency of risk-sharing Welfare effects of regulation In our model, we can interpret the assets with low collateral-requirement as a subprime loan. In particular they carry higher interest rates, and tend to be bought by agents who lack collateralizable durable goods in the present. Should one banish subprime loans? We find out that restricting trade in the subprime assets tends to hurt all agents. This was a robust feature in our numerical investigations of the model. On the other hand, in some cases, both rich and poor agents gain if only subprime loans can be traded (and markets for prime loans are shut down). However, the middle-class loses if only subprime loans can be traded and it is therefore not Pareto-improving. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

41 Economy Introduction The model Numerical examples Conclusion Economy Utility maximization problem Some theoretical observations We consider a pure exchange economy over two periods t = 0, 1. s S = {1,..., S} : set of states in period 1; S = S + 1 : set of all states; l L = {1, 2} : set of commodities or goods; Y s = (0, 1) for each state s : consumption-durability technology; h H = {1,... H} : set of agents; e h R S L + : initial endowment of agent h; u h : R S L + R : utility function of agent h; j J = {1,..., J} : set of assets; A j R SL + : promise per unit of asset j of each good l L in each state s S. We assume that A j = (1, 0) T in period 1. C j R L + : borrower collateral requirement. We will assume that: S = J Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

42 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Example 1: Plentiful collateralizable goods can lead to the Arrow-Debreu allocation Two states in period 1 S = {0, 1, 2}; Two agents, H = {1, 2} with utility function of the form: u h =0.2 log(x 1 (0)) log(x 2 (0)) (0.2 log(x 1 (s)) log(x 2 (s))) 2 s=1 Suppose that durable goods is plentiful and endowments are: e 1 (0) = (4, 2), e 1 (1) = (4, 0), e 1 (2) = (4, 0); e 2 (0) = (2, 2), e 2 (1) = (6, 0), e 2 (2) = (2, 0). Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

43 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Example 1: Plentiful collateralizable goods can lead to the Arrow-Debreu allocation The set {C j, j J CC } = {p 1 (s)/p 2 (s), s S} consists of the two assets with collateral requirements C 1 = 0.1 and C 2 = With two states, these two assets are sufficient to complete the markets. The crucial point of this example is that each agent has so much collateralizable goods that collateral constraints are not binding and agents can trade to the complete markets allocation. Agent 1 portfolio zj 1 = θ1 j φ1 j is given by z1 1 = 5.2 and z2 1 = 4. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

44 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Example 1: Unequal distribution of collateralizable goods affects risk-sharing Now, we assume that endowments are: e 1 (0) = (4, 4), e 1 (1) = (4, 0), e 1 (2) = (4, 0); e 2 (0) = (2, 0), e 2 (1) = (6, 0), e 2 (2) = (2, 0). Agent 1 s portfolio is now z1 1 = and z1 2 = 0 and second period risk is not shared at all. We report welfare numbers in terms of wealth equivalence compared to the Arrow-Debreu allocation. We compute W R h = exp( uhgeicc u had 2 ). The welfare rates in this example are: (W R 1, W R 2 ) = (0.9998, ). Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

45 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Example 2: With scarce collateralizable goods only few assets are traded Four states in period 1 S = {0, 1,..., 4}. Two agents, H = {1, 2}, each with identical utility, u h (x) = log(x 1 (0)) + log(x 2 (0)) (log(x 1 (s)) + log(x 2 (s))) 4 We consider a variety of profiles of endowments, differing by distribution of durable (collateralizable) good in the first period: s=1 e 1 (0) = (4, η), e 1 (1) = e 1 (2) = (1, 0), e 1 (3) = e 1 (4) = (2, 0); e 2 (0) = (1, (1 η)), e 2 (1) = e 2 (3) = (1, 0), e 2 (2) = e 2 (4) = (2, 0.2). We consider η 1/2 Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

46 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Since we assume identical homothetic utility, spot-prices do not depend on η (distribution of durable good in the first period). The set {C j, j J CC } = {p 1 (s)/p 2 (s), s S} consists of the four assets with collateral requirements C 1 = 0.5, C 2 = 0.4, C 3 = and C 4 = 0.3. The assets payment in the states is defined by follows in Table below: min{p 1 (s),p 2 (s)c j } p 1 (s) Table: Assets payment in the states Assets state 1 state 2 state 3 state 4 j= j= j= j= Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

47 Introduction The model Numerical examples Conclusion Example 2: Only few assets are traded Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Table: Portfolio agent 1 for different values η (agent 1 s durable good in the first period) η asset 1 asset 2 asset 3 asset The only asset traded is the one with the lowest margin requirement. Asset 2 is traded for risk-sharing in the second period. Agent 2 has sufficient collateral to sell only asset 2. Buying asset 3 is a way for the borrower to insure. Both agents have sufficient collateral to establish short positions. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

48 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Example 2: How large are the welfare losses? Table: Welfare rate for distribution of durable good: agent 1 and 2 η Lender (agent 1) Borrower (agent 2) Agent 2 would gain more than 7 percent if he could commit to pay back all promises and trade in all assets without holding any collateral. Between η = 0.8 and η = 0.5 the welfare losses remain more or less constant for agent 1, while there are still substantial improvements for agent 2. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

49 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Example 2: Exogenously selecting margin requirements can make both agents better off? In this example all agents have identical homothetic utility. According to the identical homothetic utility Theorem is impossible to make both agents better off, by exogenously selecting margin requirements. This obviously does not imply, however, that all possible margin-requirements are Pareto-ranked. First we assume η = 0.95, i.e, e 1 2(0) = 0.95, e 2 2(0) = 0.05 In this case, it seems likely that GEIC equilibrium allocations are in fact Pareto-ranked. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

50 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Figure: Welfare Rate for regulated collateral (η = 0.95) The highest utility for both agents is with low collateral (red point), this is true for all η Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

51 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example For the case η = 0.85, i.e., e 1 2 (0) = 0.85, e2 2 (0) = 0.15 the GEIC equilibria are not Pareto-ranked. Now low collateral is good only lender (red point), this is true for all 0.82 η Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

52 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Example 3: Three agents with heterogeneous utility The three agents endowments are given by: e 1 (0) = (4, η), e 1 (1) = (1, 0), e 1 (2) = (4, 0), e 1 (3) = (2, 0); e 2 (0) = (1, γ), e 2 (1) = (1, 0), e 2 (2) = (2, 0), e 2 (3) = (4, 0); e 3 (0) = (2, 1 η γ), e 3 (1) = (2, 0.2), e 3 (2) = (2, 0), e 3 (3) = (2, 0.2). We assume that there are S=3 states in the second period; We assume that agents have heterogeneous utility with u h (x) =α h log(x 1 (0)) + (1 α h ) log(x 2 (0)) (α h log(x 1 (s)) + (1 α h ) log(x 2 (s))) 3 s=1 α 1 = 0.7, α 2 = 0.77, α 3 = Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

53 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Example 3: Regulating collateral-requirement Since preferences are heterogeneous, identical homothetic utility Theorem no longer applies; To investigate this question, we search for GEIRC equilibria that could be Pareto better; We examine the case η = 0.8, γ = 0.1, i.e., e 1 2(0) = 0.80, e 2 2(0) = 0.10, e 3 2(0) = 0.10 While it is still impossible to Pareto-improve on the GEICC allocation, both agents 2 and 3 can obtain relatively large gains through a regulation. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

54 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Figure: Welfare rate for regulated collateral (e 1 2 (0) = 0.80, e2 2 (0) = 0.10, e3 2 (0) = 0.10) GEIRC3 GEICC GEIRC agent GEIRC agent 2 Agents 2 and 3 can obtain gains through a regulation (GEIRC1). Low collateral is good for lender (red point - GEIRC2). Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

55 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Example 4: Calibration to match real world data We want to investigate the role of sub-prime loans for risk-sharing as well as who in the economy gains and who loses through regulation; We assume that there are 4 types of agents whose endowments we calibrated to match the income and wealth distribution in US data; We interpret endowments in the non-durable as income while endowments in the durable good are interpreted as wealth; The estimates on wealth and income distribution in the US for 1995 and 2004 follow Di, Z.X. (2007). Growing Wealth, Inequality, and Housing in the United States. Harvard University s Joint Center for Housing Studies W07-1. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

56 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example Four states in period 1 S = {0, 1,..., 4}. Four types of agents, H = {1,..., 4}. We assume that agents have heterogeneous utility: u h (x) =α h log(x 1 (0)) + (1 α h ) log(x 2 (0)) 4 + ε s (α h log(x 1 (s)) + (1 α h ) log(x 2 (s))) s=1 We choose α h to roughly match a relative price of durable to non-durable good of 1/2 α 1 = 0.5, α 2 = 0.4, α 3 = 0.3, α 4 = 0.6 Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

57 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example The four agents endowments are given by: e 1 (0) = (0.61, 0.84), e 1 (1) = e 1 (3) = (0.63, 0), e 1 (2) = e 1 (4) = (0.21, 0); e 2 (0) = (0.22, 0.12), e 2 (1) = e 2 (3) = (0.21, 0), e 2 (2) = e 2 (4) = (0.63, 0); e 3 (0) = (0.12, 0.04), e 3 (1) = e 3 (2) = (0.11, 0), e 3 (3) = e 3 (4) = (0.05, 0); e 4 (0) = (0.05, 0.00), e 3 (1) = e 3 (2) = (0.05, 0), e 3 (3) = e 3 (4) = (0.11, 0). Probabilities are given by: ε s = (0.60, 0.18, 0.18, 0.04). Since preferences are heterogeneous, we search for GEIRC equilibria that could be Pareto better. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

58 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example In the GEICC equilibria two assets are traded. GEIRC1 corresponds to the equilibrium where there is full default for all traded assets. GEIRC2 corresponds to the equilibrium with default occurring only for asset 4 in the states 1 and 3 (i. e. in this economy the sub-prime loans are not available). GEIRC3 corresponds to the equilibrium where all agents to trade in an asset that never defaults. Table: Portfolios Agents 1, 2, 3 and 4 GEIC Agent 1 Agent 2 Agent 3 Agent 4 CC (0,0.37,-0.16,0) (0,-0.24,0.11,0) (0,-0.11,0.05,0) (0,-0.02,0,0) RC1 (0,0.43,-0.09,0) (0,-0.21,0,0) (0,-0.19,0.09,0) (0,-0.03,0,0) RC2 (0,-0.16,0,0.34) (0,0.11,0,-0.23) (0,0.05,0,-0.10) (0,0,0,-0.01) RC3 (0.11,0.06,0,0) (-0.11,0,0,0) (0,-0.05,0,0) (0,-0.01,0,0) Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

59 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example agent GEICC GEIRC GEIRC GEIRC agent 1 In the GEICC equilibria two assets are traded. The rich agent 1 lends in the sub-prime asset (0.37 units) and borrows (0.16 units) in the safe asset. Agents 2-4 borrow exclusively sub-prime, while agents 2 and 3 (the middle-class) actually saves some money in the safe bond. GEIRC1 corresponds to the equilibrium where there is full default for all traded assets. The rich agent 1 benefits from lending more units in the subprime asset (0.43 units), due to higher interest rate, in relation to GEICC equilibrium. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

60 Introduction The model Numerical examples Conclusion Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example GEICC GEICC GEIRC GEIRC GEIRC2 agent GEIRC1 agent GEIRC agent 2 GEIRC agent 1 Agents 2 and 3 cannot be made better off through any regulation (GEICC point). Agents 1 and 4 gain simultaneously if only subprime borrowing is allowed (GEIRC1 point). Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

61 Introduction The model Numerical examples Conclusion Example 4: Robustness analysis Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example In order to verify if the previous specification are robust we consider more specifications for preferences. In Yao, R. and Zhang, H.H. (2005). Optimal Consumption and Portfolio Choices with Risky Housing and Borrowing Constraints. The Review of Financial Studies preferences over housing and other goods are represented by the Cobb-Douglas function. They estimate the housing preference (1 α h ) as 0.2 for US in 2001 based on the average proportion of household housing expenditure according to the Bureau of Labor Statistics (BLS) of the US Department of Labor. Here we estimate the durable good preference (1 α h ) for each type of agent, based on the proportions of housing, furniture and vehicle purchases according to the BLS for Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

62 Introduction The model Numerical examples Conclusion Example 4: Robustness analysis Example 1: Plentiful durable good can lead to complete markets Example 2: Scarce durable goods markets appear incomplete Example 3: Regulating collateral with heterogeneous utility Example 4: A calibrated example In this case the preference are: α 1 = 0.74, α 2 = 0.73, α 3 = 0.72, α 4 = 0.71 As in the above example, both agents 1 and 4 can be made better off when trade is restricted to be in the sub-prime asset. The following values for α also give this result: α 1 = 0.7, α 2 = 0.6, α 3 = 0.5, α 4 = 0.4 α 1 = 0.4, α 2 = 0.3, α 3 = 0.5, α 4 = 0.7 α 1 = 0.5, α 2 = 0.4, α 3 = 0.3, α 4 = 0.2 indicating that what seems to be the most robust case is a case as in the above example. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete

63 Conclusion Introduction The model Numerical examples Conclusion Show that in the GEIC equilibrium generally only few of the possible contracts are traded with scarce and unequal distribution of durable (collateralizable) good; A regulation of collateral requirements never leads to a Pareto-improvement for all agents. However, we show that equilibria corresponding to different regulated collateral requirements are often not Pareto-ranked, and some agents can be benefited with regulation; In our numerical examples, it is never optimal to regulate the market for subprime loans (middle-class loses from such a regulation). However, in all cases the subprime asset is traded actively and provide the only possibility for the poor agent to purchase the durable good. Aloísio Araújo, Felix Kubler and Susan Schommer Regulating collateral-requirements when markets are incomplete