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Annals of Finance

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01.11.2012 | Research Article

On Ponzi schemes in infinite horizon collateralized economies with default penalties

verfasst von: V. Filipe Martins-da-Rocha, Yiannis Vailakis

Erschienen in: Annals of Finance | Ausgabe 4/2012

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Abstract

We show, by means of an example, that in models where default is subject to both collateral repossession and utility punishments, opportunities for doing Ponzi schemes are not always ruled out and (refined) equilibria may fail to exist. This is true even if default penalties are moderate as defined in Páscoa and Seghir (Game Econ Behav 65:270–286, 2009). In our example, asset promises and default penalties are chosen such that, if an equilibrium does exist, agents never default on their promises. At the same time collateral bundles and utility functions are such that the full repayment of debts implies that the asset price should be strictly larger than the cost of collateral requirements. This is sufficient to induce agents to run Ponzi schemes and destroy equilibrium existence.

Vorheriger Artikel More punishment, less default?

Nächster Artikel A two price theory of financial equilibrium with risk management implications

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In that respect, we provide a counterexample to the existence result claimed in Páscoa and Seghir (2009). If the arguments in the Proof of Theorem 4.1 in Páscoa and Seghir (2009) were correct, then we would get existence of a refined equilibrium when default penalties are moderate.

In other words, we show that moderate penalties can be an effective mechanism (as defined in Ferreira and Torres-Martínez (2010)).

This is not true for equilibria that are not refined since lenders may expect the asset to deliver nothing above the depreciated value of the collateral, despite the fact that default penalties would induce agents to repay fully their debt in case of trade.

The unitary default penalty $\mu ^i_t$ represents the instantaneous disutility from defaulting in real terms the market value of the bundle $w_t$.

One of the equilibrium conditions will require that lenders’ expected return $V_t(\kappa ,p)$ coincides with the actual deliveries of the borrowers in the sense that

$$\begin{aligned} \sum _{i\in I} V_t(\kappa ,p) \theta ^i_{t-1} = \sum _{i\in I} d^i_t. \end{aligned}$$

By convention we let $a_{-1}=(x_{-1},\theta _{-1},\varphi _{-1},d_{-1})=(0,0,0,0)$.

This issue is ignored by Páscoa and Seghir (2009).

By convention $x^i_{-1}=0$ and $Y_{-1} = 0$.

This assumption is automatically satisfied if the sequence of functions $(v^i_t)_{t\in \mathcal T }$ is uniformly bounded from above by an increasing function $\overline{v}^i$. Indeed, in that case we have

$$\begin{aligned} U^i(\varOmega ) \leqslant \overline{v}^i(\overline{\varOmega }) \sum _{t\geqslant 0} [\beta _i]^t < \infty \end{aligned}$$

where $\overline{\varOmega } = \sum _{i\in I} \overline{\varOmega }^i$.

If the promise bundle $A_t$ and the depreciated collateral bundle $Y_t C_{t-1}$ are not zero then $D_t(p)$ (and consequently $V_t(p)$) are not zero since $p_t$ is strictly positive.

See Dubey et al. (2005) and the discussion in Martins-da Rocha and Vailakis (2012).

By convention we let $a_{-1}=(x_{-1},\theta _{-1},\varphi _{-1},d_{-1})=(0,0,0,0)$ and $b_0(\kappa ,p)=0$.

Páscoa and Seghir (2009) assumed that default penalties are moderate and claimed in Theorem 4.1 that an equilibrium exists. Actually, the only difficult step (which is also the only step where the assumption of moderate penalties is used) of their proof consists of proving that if a sequence of finite horizon equilibria converges then for every agent, the limiting plan is optimal for the infinite horizon budget set. If their arguments were correct we would also get existence of an $\varepsilon $-equilibrium when default penalties are moderate since optimality of a plan among budget feasible plans is independent of whether we consider equilibria or $\varepsilon $-equilibria (individual demand sets coincide for both concepts).

The mistake in the intuitive argument we provide (and in the proof proposed by Páscoa and Seghir (2009)) is that when contemplating an alternative budget feasible plan, an agent does not restrict his choices to be physically feasible. In particular, depending on the sequence of prices, a budget feasible plan may have a sequence of asset short-sales that is inconsistent with the scarcity of goods (recall that when short-selling an agent should constitute collateral in terms of goods).

The set $\Delta (L)$ is the simplex in $\mathbb R ^L_+$, i.e., $\Delta (L)=\{ p \in \mathbb R ^L_+ \ :\ \sum _{\ell \in L} p(\ell ) =1\}$.

Observe that the term $b_t(\kappa ,p)$ appearing in the market clearing condition (3.1) satisfies $b_t(\kappa ,p) \leqslant \overline{b}_t$.

The fact that default penalties are moderate is used in Claim 4.1 which is essential in order to get condition (d) in the Theorem of Appendix .

We thank Juan Pablo Torres–Martínez for pointing out that this issue is delicate and deserves some attention.

One should apply the theorem in Appendix by choosing $L_t = L \cup \{1,2,3\}$ or equivalently $\mathbb R ^{L_t}=\mathbb R ^L \times \mathbb R ^3$. Condition (L.3) in Appendix follows from Assumptions (A.1) and (A.2). Condition (b) in the theorem of Appendix follows from Remark 2.1. For more details, we refer to Appendix .

We let $\rho ^i_0=0$ since there is no delivery at the initial period $t=0$.

By convention, we let $a_{-1}=(0,0,0,0)$.

See Statement 2 in the theorem of Appendix .

Observe that for any period $t<\tau $, we have $\mathcal L ^i_t(a_t,a_{t-1})=\mathcal L ^i_t(a^i_t,a^i_{t-1}) = \varPi _t^i(a^i_t,a^i_{t-1})$; and for any period $t>\tau +1$ we have $\mathcal L ^i_t(a_t,a_{t-1}) = \mathcal L ^i_t(0,0) \geqslant 0$.

We borrowed from Páscoa and Seghir (2009) the idea that we can choose exogenously the default penalty such that, endogenously at equilibrium, no agent will decide to default. This is possible due to the bound on marginal wealth obtained in Proposition 4.2. We only succeeded to find such a bound when default penalties are moderate. In particular, we do not know for the examples proposed by Páscoa and Seghir (2009) whether Ponzi schemes reappear when unduly pessimistic expectations on asset deliveries are ruled out.

If we do not consider an $\varepsilon $-equilibrium, then one may have $\kappa _t <1$ and no trade in period $t$. In that case, our argument does not apply.

In that respect, we show that the arguments in (Páscoa and Seghir (2009), Theorem 4.1) are not correct.

As usual $\nabla v^i(x) = (\nabla v^i_\ell (x(\ell )), \nabla v^i_g(x(g)))$ is the gradient of $v^i$ at $x$ where $\nabla v^i_{\ell }$ and $\nabla v^i_g$ are the differential of $v^i_\ell $ and $v^i_g$ respectively.

Observe that if we replace $\omega _t$ by $\underline{\omega }$ in the definition of $b_t$ given in Proposition 4.3 then we get $\underline{b}$. In particular we have $b_t \geqslant \underline{b}$ for every $t\geqslant 1$.

Recall that $b_{t+1}/\underline{b} = v^i(\omega ^i_t)/v^i(\underline{\omega }) \geqslant 1$.

This in particular shows that the sufficient condition proposed by Ferreira and Torres-Martínez (2010) is not necessary.

The arguments can easily be adapted to prove existence of an $\varepsilon $-equilibrium and then of a refined equilibrium.

The extension to a model with uncertainty and incomplete markets is only a matter of notation.

If $f:\mathbb R ^n \rightarrow [-\infty ,\infty )$ is a concave function with $f(x) >-\infty $, then the derivative of $f$ at $x$ in the direction of $y\in \mathbb R ^n$ is

$$\begin{aligned} df(x;y) = \lim _{\lambda \downarrow 0} \frac{f(x+\lambda y) - f(x)}{\lambda }. \end{aligned}$$

We must have $p_t\mathbf 1 _L >0$ since $v^i_t$ is strictly increasing.

Observe that

$$\begin{aligned} \lim _{\varepsilon \downarrow 0} \frac{v^i_t(x^i_t + \varepsilon (C_t + \alpha \mathbf 1 _L)) - v^i_t(x^i_t)}{\varepsilon } = dv^i_t(x^i_t;C_t + \alpha \mathbf 1 _L) > dv^i_t(x^i_t;C_t). \end{aligned}$$

Condition (A.2) is satisfied if

$$\begin{aligned} \forall i\in I, \quad \inf _{y\in [0,\varOmega _t]} d v^i_t(y;C_t) \geqslant \beta _i \mu ^i_{t+1} \overline{b}_{t+1}. \end{aligned}$$

Observe that by concavity we have $d v^i_t(y;C_t) \geqslant v^i_t(y+C_t)-v^i_t(y)$. Since $[0,\varOmega _t]$ is a compact set and $v^i_t$ is continuous, there exists $y^i_t \in [0,\varOmega _t]$ such that

$$\begin{aligned} \inf _{y\in [0,\varOmega _t]} d v^i_t(y;C_t) \geqslant v^i_t(y^i_t+C_t)-v^i_t(y^i_t). \end{aligned}$$

Since the function $v^i_t$ is strictly increasing, we have $v^i_t(y^i_t+C_t) > v^i_t(y^i_t)$ implying that the infimum is not $0$. It follows that Condition (A.2) can be satisfied for strictly positive default penalties.

In the economic model of this paper, an action $a_t$ is a vector $(x_t,\theta _t,\varphi _t,d_t)$ called a plan where $x_t \in \mathbb R ^L$, $\theta _t \in \mathbb R $, $\varphi _t \in \mathbb R $, and $d_t \in \mathbb R $. For this case, we have $L_t = L \cup \{1,2,3\}$.

In the economic model of this paper, the constraints are the solvency constraint, the collateral requirement, the minimum delivery constraint and non-negativity constraints. For this case, we have $K_t=\{1,2,3,4\}\cup L$.

Since the domain $\text{ dom}(g_t)$ of the function $g_t$ is the whole space $\mathbb R ^{L_t}\times \mathbb R ^{L_{t-1}}$, concavity already implies that $g_t$ is continuous.

We denote by $\text{ dom}(f_t)$ the set of all points $(c_1,c_2)\in \mathbb R ^{L_t} \times \mathbb R ^{L_{t-1}}$ such that $f_{t}(c_1,c_2)\in \mathbb R $. Then, continuity is in the sense that $\widehat{f}_t : \text{ dom}(f_t) \rightarrow \mathbb R $ defined by $\widehat{f}_t(c_1,c_2) = f_t(c_1,c_2)$ is continuous on $\text{ dom}(f_t)$.

Under (a)–(d) we obtain for every finite-horizon sequence $c\in C^\tau (L)$,

$$\begin{aligned} \sum _{0 \leqslant t \leqslant {\tau +1}} \mathcal L _t(c_t,c_{t-1}) \leqslant \sum _{t \in \mathcal T } \mathcal L _t(c^\star _{t},c^\star _{t-1}) = \sum _{t \in \mathcal T } f_t(c^\star _{t},c^\star _{t-1}). \end{aligned}$$

If $\nabla \mathcal L _t^\star $ is a super-gradient of $\mathcal L _t$ at $(c^\star _t,c^\star _{t-1})$ there exist two vectors $\nabla _1 \mathcal L _t^\star \in \mathbb R ^{L_t}$ and $\nabla _2 \mathcal L _t^\star \in \mathbb R ^{L_{t-1}}$ such that

$$\begin{aligned} \mathcal L _t(\widetilde{c}_t,\widetilde{c}_{t-1}) - \mathcal L _t(c^\star _t,c^\star _{t-1}) \leqslant \nabla _1 \mathcal L _t^\star \times (\widetilde{c}_t - c^\star _t) + \nabla _2 \mathcal L _t^\star \times (\widetilde{c}_{t-1} - c^\star _{t-1}) \end{aligned}$$

for every pair $(\widetilde{c}_t,\widetilde{c}_{t-1})$ in $\mathbb R ^{L_t} \times \mathbb R ^{L_{t-1}}$. The super-gradient $\nabla \mathcal L _t^\star $ is then assimilated with the pair $(\nabla _1 \mathcal L _t^\star ,\nabla _2 \mathcal L _t^\star )$. Observe that $\nabla _1 \mathcal L _t^\star $ belongs to the super-differential of the function $x\mapsto \mathcal L _t(x,c^\star _{t-1})$ at $c^\star _t$ and $\nabla _2 \mathcal L _t^\star $ belongs to the super-differential of the function $x\mapsto \mathcal L _t(c^\star _{t},x)$ at $c^\star _{t-1}$.

Observe that the horizon of the sequence $c$ is $\tau $. Since $T > \tau $, it follows that $c$ also belongs to $C^T(L)$. From Eq. (B.5) we get

$$\begin{aligned} f(c) + \sum _{0\leqslant t \leqslant T+1} \varPsi _t^T \cdot g_t(c_t,c_{t-1}) \leqslant f(c^\star ). \end{aligned}$$

From Assumption (L.1) we know that $g_ t(c_t,c_{t-1}) = g_t(0,0)\geqslant 0$ for every $t>\tau +1$. Therefore we get

$$\begin{aligned} f(c) + \sum _{0\leqslant t \leqslant \tau +1} \varPsi _t^T \cdot g_t(c_t,c_{t-1}) \leqslant f(c) + \sum _{0\leqslant t \leqslant T+1} \varPsi _t^T \cdot g_t(c_t,c_{t-1}) \leqslant f(c^\star ). \end{aligned}$$

For simplicity, the sum

$$\begin{aligned} \sum _{k\in K_t} \varPsi _{T,k} \nabla _1 g_{T,k}^\star \end{aligned}$$

is denoted by $\varPsi _T \star \nabla _1 g_T^\star $.

Replacing “$\liminf $” by “$\limsup $” in Eq. (B.11) we get $\limsup _{T \rightarrow \infty } f^T(c) \leqslant f(c^\star )$.

Observe that from Eq. (B.12) we have $\gamma _t p_t \geqslant \beta ^t \nabla v_t \gg 0$ implying that $\gamma _t>0$.

Araujo, A.P., Páscoa, M.R., Torres-Martínez, J.P.: Collateral avoids Ponzi schemes in incomplete markets. Econometrica 70, 1613–1638 (2002)CrossRef

Dubey, P., Geanakoplos, J., Shubik, M.: Default and punishment in general equilibrium. Econometrica 73, 1–37 (2005)CrossRef

Ferreira, T.R.T., Torres-Martínez, J.P.: The impossibility of effective enforcement mechanisms in collate- ralized credit markets. J Math Econ 46, 332–342 (2010)CrossRef

Kubler, F., Schmedders, K.: Stationary equilibria in asset-pricing models with incomplete markets and collateral. Econometrica 71, 1767–1793 (2003)CrossRef

Martins-da Rocha, V.F., Vailakis, Y.: Harsh default penalties lead to Ponzi schemes: a counterexample. Game Econ Behav 75, 277–282 (2012)CrossRef

Páscoa, M.R., Seghir, A.: Harsh default penalties lead to Ponzi schemes. Game Econ Behav 65, 270–286 (2009)CrossRef

Páscoa, M.R., Seghir, A.: Collateralized borrowing: direct (dis)utility effects hamper long-run equilibrium. Universidade Nova de Lisboa, Mimeo (2011)

Titel: On Ponzi schemes in infinite horizon collateralized economies with default penalties
verfasst von: V. Filipe Martins-da-Rocha
Yiannis Vailakis
Publikationsdatum: 01.11.2012
Verlag: Springer-Verlag
Erschienen in: Annals of Finance / Ausgabe 4/2012
Print ISSN: 1614-2446
Elektronische ISSN: 1614-2454
DOI: https://doi.org/10.1007/s10436-012-0209-y

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