On Ponzi schemes in infinite horizon collateralized economies with default penalties

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.

Appendices

Appendix A: Sufficient conditions on primitives to rule out Ponzi schemes

In this section we exhibit a condition relating the marginal utility of consuming the collateral bundle and the marginal penalty of default which ensures that the limits of equilibria of truncated economies are competitive equilibria of the infinite horizon economy.³¹ Our objective is neither to provide a general existence result nor to give a rigorous proof.³² We only intend to give a sketch of the proof to illustrate the intuition behind this condition.

Fix an horizon $T>1$ and consider a competitive equilibrium $(\pi ,\varvec{a})$ of the economy truncated at period $T$ where $\pi =(p,q,\kappa )$ and $a^i=(x^i,\theta ^i,\varphi ^i,d^i)$. Fix an agent $i$ and a period $t <T$. Assume that the following inequality is satisfied

$$\begin{aligned} \exists i\in I, \quad dv^i_t(x^i_t;C_t) \geqslant \beta _i \mu ^i_{t+1} \overline{b}_{t+1} \end{aligned}$$

(6.1)

where $d v^i_t(x^i_t;C_t)$ is the derivative of $v^i_t$ at $x^i_t$ in the direction of $C_t$,³³ and $\overline{b}_{t+1}$ is the maximum default in real terms at date $t+1$ defined by

$$\begin{aligned} \overline{b}_{t+1} = \sup _{p\in \Delta (L)} \frac{[pA_{t+1} - pY_{t+1}C_t]^+}{p w_{t+1}}. \end{aligned}$$

We claim that we must have $q_t \leqslant p_t C_t$. Indeed, assume by contradiction that there exists $\alpha >0$ such $q_t = p_t C_t + \alpha p_t \mathbf 1 _L$.³⁴ Suppose agent $i$ considers deviating from $\varphi ^i_t$ by “shorting” $\varepsilon$ more security, using the receipt $q_t\varepsilon$ to buy $\varepsilon$ of the collateral and $\alpha \varepsilon$ of the vector $\mathbf 1 _L$ (i.e., $\alpha \varepsilon$ units of each good). This strategy is budget feasible at date $t$, but also at date $t+1$. Indeed, agent $i$ may decide to pay back $D_{t+1}(p)\varepsilon$ which is smaller than the value of depreciated collateral $Y_{t+1}C_t \varepsilon$ constituted at period $t$. The decrease in utility at date $t$ due to default penalties at date $t+1$ is

$$\begin{aligned} \varepsilon \beta _i \mu ^{i}_{t+1} \frac{[p_{t+1} A_{t+1} - p_{t+1}Y_{t+1}C_t]^+}{p_{t+1} w_{t+1}} \leqslant \varepsilon \beta _i \mu ^i_{t+1} \overline{b}_{t+1}. \end{aligned}$$

Inequality (A.1) implies that the deviation strictly increases agent $i$’s utility,³⁵ contradicting that $a^i$ is optimal.

Inequality (A.1) depends on the consumption allocation $(x^i_t)_{i\in I}$ which is an endogenous variable. It is possible to exhibit a stronger condition only in terms of primitives. Indeed, consider the following property:

$$\begin{aligned} \forall \varvec{z}_t \in \text{ F}_t, \quad \exists i\in I, \quad d v^i_t(z^i_t;C_t) \geqslant \beta _i \mu ^i_{t+1} \overline{b}_{t+1} \end{aligned}$$

(6.2)

where $\text{ F}_t$ is the set of consumption bundles $\varvec{z}_t=(z^i_t)_{i\in I}$ physically feasible at date $t$, i.e.,

$$\begin{aligned} \sum _{i\in I} z^i_t = \varOmega _t. \end{aligned}$$

Under Eq. (A.2) condition (A.1) is automatically satisfied since $x^i_t$ is physically feasible for any competitive equilibrium.³⁶

Consider now that there is an infinite set $\text{ Barr} \subset \mathcal T$ such that Eq. (A.2) is satisfied for every date $t\in \text{ Barr}$. Let $(\pi ^T,\varvec{a}^T)$ be a competitive equilibrium for the truncated economy $\mathcal E ^T$ with finite-horizon $T$. Observe that for every $t\in \text{ Barr}$ with $t<T$, we must have $q^T_t \leqslant p_t^T C_t$. Following the (standard) arguments in Páscoa and Seghir (2009), passing to a subsequence if necessary, we can assume that the sequence $(\pi ^T,\varvec{a}^T)$ converges to some $(\pi ,\varvec{a})$ such that all markets clear and each action $a^i$ is budget feasible and optimal among finite-horizon actions in the budget set $B^i(\pi )$. The difficulty is to show that $a^i$ is optimal among budget feasible infinite horizon actions. Observe that for every $t\in \text{ Barr}$ one must have $q_t \leqslant p_t C_t$. If the utility function $U^i(x)$ is bounded for every consumption sequence $x\in X$, then it is easy to show that $\varvec{a}$ is optimal among all feasible plans. We can then conclude that $(\pi ,\varvec{a})$ is a competitive equilibrium.

Appendix B: Necessary and sufficient conditions for the existence of Langrange multipliers

In this appendix we consider an abstract infinite dimensional maximization problem and we present some necessary and sufficient conditions for optimality. Necessary conditions for optimality by means of Lagrange multipliers are first presented (see Sect. ). We then provide sufficient conditions in Sect. . In the last section we restate the previous results for the specific economic model of the associated paper.

For each period $t \in \mathcal T$, we fix a finite set $L_t$ of “types of action”.³⁷ We denote by $C(L)$ the space of all sequencees $c=(c_t)_{t \in \mathcal T }$ where $c_t$ is a vector in $\mathbb R ^{L_t}$. By convention, we pose $L_{-1}=\{1\}$ and $c_{-1}=0$ for any sequence $c \in C(L)$. For each period $T\geqslant 1$, we let $C^T(L)$ be the subset of $C(L)$ defined by

$$\begin{aligned} C^T(L) = \{ c \in C(L)\ :\ \forall t \in \mathcal T , \quad t > T \Rightarrow c_t=0 \}. \end{aligned}$$

Fixing a finite set $K_t$ of “constraints” on actions³⁸ we can define in a similar way the sets $C(K)$ and $C^T(K)$ by replacing $L_t$ by $K_t$. For period $t$, we fix an objective function $f_t : \mathbb R ^{L_t} \times \mathbb R ^{L_{t-1}} \longrightarrow \mathbb R \cup \{-\infty \}$ and a constraint function $g_t : \mathbb R ^{L_t} \times \mathbb R ^{L_{t-1}} \longrightarrow \mathbb R ^{K_t}$. For each period $T\geqslant 1$ and each sequence $c \in C(L)$, we let

$$\begin{aligned} f^T(c) = \sum _{0 \leqslant t \leqslant T} f_t(c_t,c_{t-1}). \end{aligned}$$

When the limit exists, we denote by $f(c)$ the following sum

$$\begin{aligned} f(c)= \lim _{T \rightarrow \infty } f^T(c). \end{aligned}$$

Given $c \in C(L)$, we denote by $g(c)$ the sequence in $C(K)$ defined by

$$\begin{aligned} \forall t \in \mathcal T , \quad [g(c)]_t=g_t(c_t,c_{t-1}) \quad \text{ or} \text{ equivalently} \quad g(c)=(g_t (c_t,c_{t-1}))_{t\in \mathcal T }. \end{aligned}$$

The vector $g_t(c_t,c_{t-1})$ in $\mathbb R ^{K_t}$ is denoted by $(g_{t,k}(c_t,c_{t-1}))_{k\in K_t}$ where $g_{t,k}$ is interpreted as the $k$th constraint function from $\mathbb R ^{L_t} \times \mathbb R ^{L_{t-1}}$ to $\mathbb R$.

We assume that

• (L.1) for each period $t$, the functions $f_t$ and $g_t$ are concave with $f_t(0,0) = 0$ and $g_t(0,0) \geqslant 0$;

• (L.2) for each period $t$, the function $g_t$ is continuous³⁹ and the function $f_t$, when restricted to its domain $\text{ dom}(f_t)$, is continuous;⁴⁰

• (L.3) for each period $T\geqslant 1$, there exists a sequence $\widehat{c} \in C^T(L)$ such that

  $$\begin{aligned} f(\widehat{c}) \geqslant 0, \quad g_{T+1}(0,\widehat{c}_T) \geqslant 0 \quad \text{ and} \quad \forall t \in \{0,\ldots ,T\}, \quad g_t(\widehat{c}_t,\widehat{c}_{t-1}) \in \mathbb R ^{K_t}_{++}. \end{aligned}$$

1.1 B.1 Necessary conditions

Applying sequentially a finite dimensional convex separation argument, we can prove the following result.

Theorem

Assume that there exists $c^\star \in C(L)$ such that

• (a) the sequence $c^\star$ satisfies the constraints $g(c^\star ) \geqslant 0$;

• (b) the sum $f(c^\star )$ is well defined;

• (c) the sequence $c^\star$ is optimal among finite-horizon sequences, i.e., for any period $\tau \geqslant 1$, for every finite-horizon sequence $c \in C^\tau (L)$ satisfying the constraints $g(c)\geqslant 0$, we have $f(c^\star ) \geqslant f(c)$.

Then the following properties hold.

• 1. There exists $\varPsi \in C(K)$ with $\varPsi _t \in \mathbb R ^{K_t}_+$ such that for any period $\tau \geqslant 1$ and any finite-horizon sequence $c\in C^\tau (L)$,

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

  (7.1)

  where $\mathcal L _t(c_t,c_{t-1}) = f_t(c_t,c_{t-1}) + \varPsi _t \cdot g_t(c_t,c_{t-1})$.

• 2. If moreover, we have

  • (d) for any period $t\geqslant 1$, there exist $\tau \geqslant t$ and a finite-horizon sequence $\check{c} \in C^{\tau +1}(L)$ satisfying $g(\check{c}) \geqslant 0$, $f(\check{c}) \geqslant f^{\tau }(c^\star )$ and $c^\star \mathbf 1 _{[0,\tau ]} = \check{c}\mathbf 1 _{[0,\tau ]}$

  then⁴¹

  $$\begin{aligned} \forall t \in \mathcal T , \quad \varPsi _t \cdot g_t(c^\star _{t},c^\star _{t-1})=0. \end{aligned}$$

  (7.2)

  In particular we obtain the following variational property: for every sequence $\widetilde{c}$ in $C(L)$ and every period $T\geqslant 1$ we have

  $$\begin{aligned} \sum _{t=0}^T \mathcal L _t(\widetilde{c}_t,\widetilde{c}_{t-1}) + \mathcal L _{T+1}(c^\star _{T+1},\widetilde{c}_T) \leqslant \sum _{t=0}^{T+1} \mathcal L _t(c^\star _t,c^\star _{t-1}). \end{aligned}$$

  (7.3)

  Therefore, for every period $T\geqslant 1$, there exist a family of super-gradients⁴²

  $$\begin{aligned} (\nabla \mathcal L _0^\star , \ldots , \nabla \mathcal L ^\star _T) \quad \text{ where} \quad \nabla \mathcal L _t^\star = (\nabla _1 \mathcal L _t^\star ,\nabla _2 \mathcal L _t^\star ) \in \partial \mathcal L _t(c^\star _t,c^\star _{t-1}) \end{aligned}$$

  and $\xi _{T+1}$ a super-gradient of the function $x \mapsto \mathcal L _{T+1}(c^\star _{T+1},x)$ such that

  $$\begin{aligned} \forall t \in \{0,\ldots ,T-1\}, \quad \nabla _1 \mathcal L ^\star _t + \nabla _2 \mathcal L ^\star _{t+1} = 0 \quad \text{ and} \quad \nabla _1 \mathcal L ^\star _T + \xi _{T+1} =0\qquad \quad \end{aligned}$$

  (7.4)

Proof of the theorem

Fix a period $T\geqslant 1$. We let $A$ be the subset of $\mathbb R \times C^{T+1}(K)$ defined by

$$\begin{aligned} A = \left\{ (\alpha ,d) \in \mathbb R \times C^{T+1}(K) \ :\ \exists c \in C^T(L), \quad \alpha \leqslant f(c)-f(c^\star ) \quad \text{ and} \quad d \leqslant g(c) \right\} \end{aligned}$$

and we let $B = (0,\infty ) \times C^{T+1}_+(K)$ where

$$\begin{aligned} C^{T+1}_+(K) = \prod _{0\leqslant t \le {T+1}} \mathbb R ^{K_t}_+. \end{aligned}$$

Following Assumptions (L.1)–(L.3) and conditions (a)–(c), the sets $A$ and $B$ are disjoint non-empty convex subsets of $\mathbb R \times C^{T+1}(K)$. It follows from the finite dimensional separating hyperplane theorem that there exists a non-zero pair $(\mu ^T,\varPsi ^T) \in \mathbb R _+ \times C^{T+1}_+(K)$ such that

$$\begin{aligned} \forall c \in C^T(L), \quad \mu ^T f(c) + \sum _{0 \le t \le {T+1}} \varPsi ^T_t \cdot g_t(c_t,c_{t-1}) \le \mu ^T f(c^\star ). \end{aligned}$$

(7.5)

Following Assumption (L.3), we can take $\mu ^T=1$ without any loss of generality.

Fix a period $\tau \in \mathcal T$. The objective is to prove that the sequence $(\varPsi ^T_\tau )_{T\geqslant 1}$ converges in $\mathbb R _+$. Following Assumption (L.3), there exists a process $\widehat{c} \in C^\tau (L)$ such that

$$\begin{aligned} f(\widehat{c}) \geqslant 0, \quad g_{\tau +1}(0,\widehat{c}_\tau )\geqslant 0 \quad \text{ and} \quad \forall t \in \{0,\ldots ,\tau \}, \quad \varepsilon _t \equiv g_t(\widehat{c}_t,\widehat{c}_{t-1}) \in \mathbb R ^{K_t}_{++}. \end{aligned}$$

Observe that for any $t > \tau +1$ we have $g_t(\widehat{c}_t,\widehat{c}_{t-1}) = g_t(0,0) \geqslant 0$. It follows from Eq. (B.5) that for all $k\in K_\tau$,

$$\begin{aligned} \varPsi ^T_\tau (k) \leqslant \frac{f(c^\star )}{\varepsilon _\tau (k)}. \end{aligned}$$

Using a diagonal procedure and passing to a subsequence if necessary, we can prove that there exists $\varPsi \in C_+(K)$ such that

$$\begin{aligned} \forall \tau \in \mathcal T , \quad \varPsi _\tau = \lim _{T\rightarrow \infty } \varPsi ^T_\tau . \end{aligned}$$

Now we fix a period $\tau \geqslant 1$ and a finite-horizon sequence $c \in C^\tau (L)$. For each $T> \tau$, it follows from Eq. (B.5) and Assumption (L.1) that⁴³

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

Passing to the limit when $T$ goes to infinite, we get the desired result (B.1):

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

Now assume that (d) is satisfied. Fix a period $t \in \mathcal T$, there exist $\tau \geqslant t$ and a finite-horizon sequence $\check{c} \in C^{\tau +1}(L)$ satisfying $g(\check{c}) \geqslant 0$, $f(\check{c}) \geqslant f^{\tau }(c^\star )$ and $\check{c} \mathbf 1 _{[0,\tau ]} = c^\star \mathbf 1 _{[0,\tau ]}$. Observe that $f^{\tau +2}(\check{c}) = f(\check{c}) \geqslant f^{\tau }(c^\star )$. Choosing $c=\check{c}$ in Eq. (B.1), it follows that

$$\begin{aligned} f^\tau (c^\star ) + \varPsi _t \cdot g_t(c^\star _t,c^\star _{t-1}) \leqslant f^{\tau +2}(\check{c}) + \sum _{0 \leqslant s \leqslant {\tau +2}} \varPsi _s \cdot g_s(\check{c}_s,\check{c}_{s-1}) \leqslant f(c^\star ). \end{aligned}$$

Since

$$\begin{aligned} \lim _{\tau \rightarrow \infty } f^\tau (c^\star ) = f(c^\star ) \end{aligned}$$

we get the desired result (B.2).

Now fix a sequence $\widetilde{c}$ in $C(L)$ and a period $T\geqslant 1$. For every $\tau >T$ we let $c$ be the sequence in $C^{\tau }(L)$ defined by

$$\begin{aligned} c_t = \left\{ \begin{array}{lcl} \widetilde{c}_t&\text{ if}&t \leqslant T \\ c^\star _t&\text{ if}&T+1 \leqslant t \leqslant \tau \\ 0&\text{ if}&\tau +1 \leqslant t \end{array} \right. \end{aligned}$$

It follows from Eqs. (B.1) and (B.2) that

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

Given the construction of the sequence $c$, we have

$$\begin{aligned}&\sum _{t=0}^T \{ \mathcal L _{t}(\widetilde{c}_t,\widetilde{c}_{t-1}) - \mathcal L _{t}(c^\star _t,c^\star _{t-1})\} + \{ \mathcal L _{T+1}(c^\star _{T+1},\widetilde{c}_T) - \mathcal L _{T+1}(c^\star _{T+1},c^\star _{T}) \} \\&\quad \leqslant \sum _{s \geqslant \tau +2} f_s(c^\star _s,c^\star _{s-1}). \end{aligned}$$

Passing to the limit when $\tau \rightarrow \infty$ and using the fact that the infinite sum $f(c^\star )$ is well-defined, we get the desired result

$$\begin{aligned} \sum _{t=0}^T \mathcal L _{t}(\widetilde{c}_t,\widetilde{c}_{t-1}) + \mathcal L _{T+1}(c^\star _{T+1},\widetilde{c}_T) \leqslant \sum _{t=0}^T \mathcal L (c^\star _t,c^\star _{t-1}) + \mathcal L _{T+1}(c^\star _{T+1},c^\star _{T}). \end{aligned}$$

We let $H$ be the function defined on $\mathbb R ^{L_0} \times \cdots \times \mathbb R ^{L_T}$ by

$$\begin{aligned} H(c_0,c_1,\ldots ,c_T) = \sum _{t=0}^T \mathcal L _{t}(c_t,c_{t-1}) + \mathcal L _{T+1}(c^\star _{T+1},c_T). \end{aligned}$$

This is a concave function having a global maximum at $(c^\star _0,\ldots ,c^\star _T)$. It follows that the super-differential of $H$ at $(c^\star _0,\ldots ,c^\star _T)$ is non-empty and contains $0$. Observe that

$$\begin{aligned} H(c_0,\ldots ,c_T) = \sum _{t=0}^T H_t(c_0,\ldots ,c_T) + h_{T+1}(c_0,\ldots ,c_T) \end{aligned}$$

where

$$\begin{aligned} H_{t}(c_0,\ldots ,c_T) = \mathcal L _t(c_t,c_{t-1}) \quad \text{ and} \quad h_{T+1}(c_0,\ldots ,c_T) = \mathcal L _{T+1}(c^\star _{T+1},c_T). \end{aligned}$$

It follows that

$$\begin{aligned} 0 \in \sum _{t=0}^T \partial H_t(c^\star _0,\ldots ,c^\star _T) + \partial h_{T+1}(c^\star _0,\ldots ,c^\star _T). \end{aligned}$$

(7.6)

Observe that a super-gradient in $\partial H_t(c^\star _0,\ldots ,c^\star _T)$ is a vector in $\mathbb R ^{L_0} \times \cdots \times \mathbb R ^{L_T}$ of the following form

$$\begin{aligned} (0,\ldots ,0,\underbrace{\nabla _2\mathcal L _t^\star }_{t-1},\underbrace{\nabla _1 \mathcal L _{t}^\star }_{t},0,\ldots ,0) \end{aligned}$$

(7.7)

where $(\nabla _1\mathcal L _t^\star ,\nabla _2 \mathcal L _{t}^\star )$ is a super-gradient in $\partial \mathcal L _t(c^\star _t,c^\star _{t-1})$. Observe moreover that a super-gradient in $\partial h_{T+1}(c^\star _0,\ldots ,c^\star _T)$ takes the following form

$$\begin{aligned} (0,\ldots ,0,\xi _{T+1}) \end{aligned}$$

(7.8)

where $\xi _{T+1} \in \mathbb R ^{L_t}$ is a super-gradient of the function $x \mapsto \mathcal L _{T+1}(c^\star _{T+1},x)$ at $c^\star _T$. Combining Eqs. (B.6), (B.7) and (B.8) we get the desired result (B.4). $\square$

1.2 B.2 Sufficient conditions

In this subsection, we investigate under which conditions the first-order conditions (B.4) in the theorem are sufficient to obtain optimality. Let $c^\star \in C(L)$ satisfying the conditions (a) and (b) of the theorem, i.e., the sequence $c^\star$ satisfies the constraint $g(c^\star ) \geqslant 0$ and the sum $f(c^\star )$ is well defined. Let $\varPsi \in C(K)$ be a sequence of Lagrange multipliers $\varPsi _t =(\varPsi _{t,k})_{k\in K_t} \in \mathbb R ^{K_t}_+$ such that the first-order conditions (B.4) and the binding conditions (B.2) are satisfied. Actually we will assume a stronger property: there exists a sequence of super-gradients

$$\begin{aligned} (\nabla \mathcal L ^\star _t)_{t\in \mathcal T } \quad \text{ where} \quad \nabla \mathcal L ^\star _t = (\nabla _1 \mathcal L ^\star _t,\nabla _2 \mathcal L ^\star _t) \in \partial \mathcal L _t(c^\star _t,c^\star _{t-1}) \end{aligned}$$

such that

$$\begin{aligned} \forall t\in \mathcal T , \quad \nabla _1 \mathcal L ^\star _t + \nabla _2 \mathcal L ^\star _{t+1}=0 \end{aligned}$$

(7.9)

where we recall that $\mathcal L _t=f_t + \varPsi _t \cdot g_t$.

Let $c$ be a sequence in $C(L)$ satisfying the constraints $g(c)\geqslant 0$. Given $T\geqslant 1$, we denote by $\mathcal L ^T(c)$ the Lagrangian up to period $T$ defined by

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

By concavity and definition of the super-gradients, we have

$$\begin{aligned} \mathcal L ^T(c) - \mathcal L ^T(c^\star ) \leqslant \sum _{t=0}^T \left[ \nabla _1 \mathcal L _t^\star \cdot (c_t-c^\star _t) + \nabla _2 \mathcal L _{t}^\star \cdot (c_{t-1}-c_{t-1}^\star ) \right]. \end{aligned}$$

Rearranging the previous sum we get

$$\begin{aligned} \mathcal L ^T(c) - \mathcal L ^T(c^\star ) \leqslant \nabla _1 \mathcal L _T^\star \cdot (c_T-c^\star _T) + \sum _{t=0}^{T-1} \left[ \nabla _1 \mathcal L _t^\star + \nabla _2 \mathcal L _{t+1}^\star \right] (c_t - c^\star _t). \end{aligned}$$

Using the Euler Eq. (B.9) we get

$$\begin{aligned} \mathcal L ^T(c) - \mathcal L ^T(c^\star ) \leqslant \nabla _1 \mathcal L _T^\star \cdot (c_T-c^\star _T). \end{aligned}$$

Since $\mathcal L _t = f_t + \varPsi \cdot g_t$ it follows that there exist a super-gradient $\nabla f_t^\star$ of $f_t$ at $(c^\star _t,c^\star _{t-1})$ and for each $k\in K_t$ a super-gradient $\nabla g_{t,k}^\star$ of $g_{t,k}$ at $(c^\star _t,c^\star _{t-1})$ such that

$$\begin{aligned} \nabla \mathcal L ^\star _t = \nabla f_t^\star + \sum _{k\in K_t} \varPsi _t \nabla g_{t,k}^\star . \end{aligned}$$

We can decompose the above super-gradients as follows

$$\begin{aligned} \nabla f_t^\star = (\nabla _1 f_t^\star ,\nabla _2 f_t^\star ) \quad \text{ and} \quad \nabla g_{t,k}^\star =(\nabla _1 g_{t,k}^\star ,\nabla _2 g_{t,k}^\star ) \end{aligned}$$

and $\nabla _1 \mathcal L _T^\star$ can be written as follows⁴⁴

$$\begin{aligned} \nabla _1 \mathcal L _T^\star = \nabla _1 f_T^\star + \varPsi _T \star \nabla _1 g_T^\star . \end{aligned}$$

We then get

$$\begin{aligned} \mathcal L ^T(c) - \mathcal L ^T(c^\star ) \leqslant \left[ \nabla _1 f_T^\star + \varPsi _T \star \nabla _1 g_T^\star \right] \cdot (c_T-c^\star _T). \end{aligned}$$

Since for every period $t$ we have

$$\begin{aligned} \varPsi _t \cdot g_t(c_t,c_{t-1}) \geqslant 0 \quad \text{ and} \quad \varPsi _t \cdot g_t(c^\star _t,c^\star _{t-1})=0 \end{aligned}$$

it follows that

$$\begin{aligned} f^T(c) - f^T(c^\star ) \leqslant \mathcal L ^T(c) - \mathcal L ^T(c^\star ) \leqslant \left[ \nabla _1 f_T^\star + \varPsi _T \star \nabla _1 g_T^\star \right] \cdot (c_T-c^\star _T). \end{aligned}$$

We obtain immediately the following properties:

Claim

If the sequence $c^\star$ satisfies the following transversality condition

$$\begin{aligned} \liminf _{T \rightarrow \infty } \left[ \nabla _1 f_T^\star + \varPsi _T \star \nabla _1 g_T^\star \right] \cdot (- c^\star _T) \leqslant 0 \end{aligned}$$

(7.10)

then $c^\star$ is optimal among finite-horizon sequences, i.e., if $c \in C^\tau (L)$ for some $\tau \in \mathcal T$ and satisfies $g(c)\geqslant 0$ then we have $f(c) \leqslant f(c^\star )$.

Now if $c$ is a (possibly infinite) sequence with $g(c)\geqslant 0$ and satisfying the following transversality condition

$$\begin{aligned} \liminf _{T \rightarrow \infty } \left[ \nabla _1 f_T^\star + \varPsi _T \star \nabla _1 g_T^\star \right] \cdot (c_T - c^\star _T) \leqslant 0 \end{aligned}$$

(7.11)

then we have⁴⁵

$$\begin{aligned} \liminf _{T \rightarrow \infty } f^T(c) \leqslant f(c^\star ). \end{aligned}$$

In particular, if $f(c)$ is well-defined then we get $f(c) \leqslant f(c^\star )$.

1.3 B.3 The economic model of the paper

In the infinite horizon economy of this paper, every agent solves a maximization problem that is a particular case of the abstract problem presented above. We fix a price sequence $\pi =(p_t,q_t,\kappa _t)_{t\in \mathcal T }$ and consider a generic agent without specifying the index $i$.

Let $L_t = L\cup \{1,2,3\}$ and identify the choice $c_t$ with the plan $(x_t,\theta _t,\varphi _t,d_t)$. We then have

$$\begin{aligned} f_t(c_t,c_{t-1}) = \beta ^t \widehat{v}_t(x_t) - \beta ^t \mu _t \frac{[V_t(p)\varphi _{t-1} - d_t]^+}{p_t w_t} \end{aligned}$$

where $\widehat{v}_t$ coincides with $v_t$ on $\mathbb R ^L_+$ and is identical to $-\infty$ elsewhere. From now on we abuse notations and identify $v_t$ with $\widehat{v}_t$.

Let $K_t=\{1,2,3,4\} \cup L$ and choose

$$\begin{aligned} g_{t,1}(c_t,c_{t-1}) = p_t\{\omega _t+ Y_t x_{t-1}\} - p_t x_t + q_t[\varphi _t - \theta _t] + V_t(\kappa ,p)\theta _{t-1} - d_t \end{aligned}$$

together with

$$\begin{aligned} g_{t,2}(c_t,c_{t-1}) = d_t - D_t(p)\varphi _{t-1}, \quad g_{t,3}(c_t,c_{t-1})=\theta _t, \quad g_{t,4}(c_t,c_{t-1})=\varphi _t \end{aligned}$$

and

$$\begin{aligned} g_{t,\ell }(c_t,c_{t-1}) = x_t(\ell ) - C_t(\ell ) \varphi _t. \end{aligned}$$

Observe that under the assumptions of our model, Assumptions (L.1)–(L.3) are automatically satisfied. A sequence $(\varPsi _t)_{t\in \mathcal T }$ with $\varPsi _t \in \mathbb R ^{K_t}=\mathbb R ^4 \times \mathbb R ^L$ is denoted by

$$\begin{aligned} \varPsi _t=(\gamma _t,\rho _t,\alpha _{\theta ,t},\alpha _{\varphi ,t},\chi _t) \quad \text{ where} \quad \chi _t=(\chi _t(\ell ))_{\ell \in L} \in \mathbb R ^L. \end{aligned}$$

The first order conditions (B.4) translate into the following form:

• (a) first order condition for consumption:

  $$\begin{aligned} \forall t \in \mathcal T , \quad \beta ^t \nabla v_t + \gamma _{t+1} p_{t+1} Y_{t+1} + \chi _t = \gamma _t p_t; \end{aligned}$$

  (7.12)

• (b) first order condition for asset purchases:

  $$\begin{aligned} \forall t \in \mathcal T , \quad \gamma _t q_t = \alpha _{\theta ,t} + \gamma _{t+1} V_{t+1}(\kappa ,p); \end{aligned}$$

  (7.13)

• (c) first order condition for deliveries:

  $$\begin{aligned} \forall t\geqslant 1, \quad \beta ^t \mu _t \frac{\delta _t}{p_t w_t} + \rho _t = \gamma _t; \end{aligned}$$

  (7.14)

• (d) first order conditions for asset sales:

  $$\begin{aligned} \forall t\geqslant 1, \quad \gamma _t q_t + \alpha _{\varphi ,t} \!=\! \rho _{t+1} D_{t+1}(p) \!+\! \chi _t C_t \!+\! \beta ^{t+1} \mu _{t+1} \frac{\delta _{t+1}}{p_{t+1} w_{t+1}} V_{t+1}(p);\qquad \quad \end{aligned}$$

  (7.15)

where $\nabla v_t$ is a super-gradient of $v_t$ at $x_t$ and $-\delta _t$ is a super-gradient of $\Delta \mapsto -[\Delta ]^+$ at $\Delta _t=V_t(p)\varphi _{t-1} - d_t$. The binding conditions (B.2) translate in the following form⁴⁶

$$\begin{aligned}&p_t x_ t + q_t \theta _t + d_t =p_t\{\omega _t+ Y_t x_{t-1}\} + q_t\varphi _t + V_t(\kappa ,p)\theta _{t-1}\\&\rho _t \{d_t - D_t(p)\varphi _{t-1}\}=0 \quad \text{ and} \quad \alpha _{\theta ,t}\theta _t = \alpha _{\varphi ,t} \varphi _t =0, \end{aligned}$$

and for every $\ell \in L$,

$$\begin{aligned} \chi _t(\ell )\{x^i_t(\ell ) - C_t(\ell )\varphi _t\}=0. \end{aligned}$$