2.1. General setup

The economy is populated by $N$ “yeoman farmer” sectors, each containing infinite consumer-producer units. Each unit in any sector produces a single type of good but may consume goods from different sectors. Indices are assumed to respect a modulo- $N$ arithmetic in the space of integers, in the sense that sector $1$ is identified with sector $N+1$ , sector $N-1$ is identified with sector $-1$ and so on. Utility of an agent of sector $j=1,\dots,N$ is given by the function

\begin{equation*} \sum _{t=0}^\infty \beta ^j U^j(t) \end{equation*}

where $0\lt \beta \lt 1$ is the intertemporal discount factor and

\begin{equation*} U^j(t)=\sum _{k\in K^j(t)}\theta _k(t)\alpha _{k-j} u\left (\frac {c^j_k(t)}{\theta _k(t)\alpha _{k-j}}\right )-v(y^j(t)) \end{equation*}

The function $u$ is continuously differentiable and satisfies the usual Inada conditions: $u'(0)=\infty$ , $u'(\infty )=0$ . $u$ is also strictly increasing and strictly concave, such that $u(0)=0$ . The function $v$ is convex, strictly increasing and continuously differentiable, and such that $v(0)=0$ .

$c^j_k(t)$ is the consumption of an agent in sector $j$ of goods produced by sector $k$ in period $t$ . $y^j(t)$ is the production of an agent in sector $j$ at period $t$ . In this extension we assume that from period $t\geq 1$ production is always fixed at the level $\bar{y}$ such that $u'(\bar{y})=v'(\bar{y})$ .Footnote ³ However, in period $t=0$ , the economy is subject to a major productivity shock described by a vector $\theta = (\theta _1(0),\ldots,\theta _N(0))$ and production will be determined by demand due to a rigid price assumption that will be explained later on. For periods $t\geq 1$ we assume $\theta _k(t) = 1$ for all sectors $k$ , and we refer to this scenario as the “normal” case.

The coefficients $\alpha _k$ define the symmetric network structure of the economy and are assumed to satisfy $\sum _k \alpha _k =1$ . $K^j(t)$ represents the subset of sectors $1,\dots,N$ , such that $\alpha _k\gt 0$ for agent $j$ in period $t$ . In a decentralized economy in the normal case when prices are equal for goods produced in different sectors, the coefficients $\alpha _{k-j}$ represent the portion of consumption that an agent in sector $j$ will dedicate to a good produced by sector $k$ . We make a symmetric specification of the model, which implies that this portion will be the same for all sectors $j = 1,\dots,N$ .Footnote ⁴ Two examples of network structures are given in Fig. 1.

Figure 1. Chain and uniform network.

Assuming equal prices for all goods, in the chain network each agent in sector $j$ demands goods only from her own sector and sector $j+1$ , with a fraction of $\lambda$ being demanded from the adjacent sector, so that $\alpha _0=1-\lambda$ and $\alpha _1=\lambda$ . In the uniform network, each agent demands the same fraction $1/N$ of consumption from each sector, with $\alpha _j=1/N$ for every $j=1,\ldots,N$ .

We focus on the effect of a productivity shock that shuts down the production by a single sector $j$ of the economy in period $t=0$ , so that $\theta _j(0) = 0$ , due, for example, to health concerns about the consumption of the good produced by this sector, or to a restriction in the production or trade of goods such as a major conflict scenario. We assume an ex ante equal probability for all of the sectors to be hit by the shock. While writing contracts and making consumption decisions, agents cannot condition any decision on the occurrence of the shock. For periods $t \geq 1$ , the economy returns to the stationary state so that the shock is completely transitory.

In this model, it is straightforward to characterize the first-best optimal allocation that would be chosen by a social planner. Since ex-ante all sectors are symmetric and subject to the same shock with equal probability, agents agree with the ranking of rotationally symmetric allocations, which are those that depend only on the occurrence of the shock $\theta$ , so that the allocation of sector $k$ when sector $j$ is hit by the negative productivity shock is the same as the allocation of sector $k+1$ when sector $j+1$ is hit by the negative productivity shock instead.

An altruistic social planner maximized the social welfare function

\begin{equation*} \sum _{t=0}^\infty \beta ^t\sum _{j=1}^N U^j(t), \end{equation*}

which is proportional to the ex-ante expected utility of each agent,Footnote ⁵ subject to the market clearing conditions in the goods market

\begin{equation*} \sum _{k \in K^j(t)} c^{j-k}_j(t)=y^j(t) \end{equation*}

for all $t\geq 0$ and $j$ . The resulting optimal allocation is such that sectors that are not hit by the negative shock will produce the optimal quantity $\bar{y}$ that satisfies $u'(\bar{y})=v'(\bar{y})$ , and consumption will be distributed such that $c^j_k(0)=\alpha _{k-j} \bar{y}$ if $\theta _k(0)=1$ and $c^j_k(0)=0$ if $k$ in the sector that is shutdown, that is, $\theta _k(0)=0$ . In the subsequent periods, all production is restored and all sectors demand the same amount, with $c^j_k(t)=\alpha _{k-j} \bar{y}$ for all $k$ .

For the rest of the paper, we evaluate the welfare consequence of different policies based on whether or not they are able to implement this first-best allocation of resources as an equilibrium. As it will be seen, while there is a fiscal policy that can implement this allocation and restore the first-best outcome, we are interested in whether or not there is a policy framework that can use a bankruptcy law as an alternative to fiscal policy since in practice implementing a complex fiscal policy is unfeasible for a government, specially if it has a restriction on the ability to raise debt or taxes, as it is in most emergent market economies.

In the decentralized economy, agents trade on a competitive market for goods and can buy government bonds that pay an interest rate in the following period that is set by the government. In each period $t$ , $p_k(t)$ is the nominal price of the good produced by sector $k$ . Prices are fixed in the previous period at a level that is assumed to make markets clear. This means that prices do not respond to a shock that is unanticipated, and production is demand determined in this situation. For period $t=0$ , prices will be determined by an initial policy trajectory that may change after the shock is learned by the agents.

Each agent begins period $t=0$ with an initial nominal wealth of $a^j(0)$ that is the result of fiscal policy. We denote by $a(t)=\sum _{j} a^j(t)$ the total government debt at period $t$ . Government bonds demanded by an agent in sector $j$ are denoted by $b^j(t)$ . Each bond pays an interest $1+i(t)$ in the following period.

Nominal wealth of agents evolve according to

\begin{equation*} a^j(t+1)=(1+i(t))b^j(t)-\tau (t+1) \end{equation*}

where $\tau (t+1)$ is a lump-sum tax collected by the government that is equal for all sectors. In the absence of private credit contracts, the budget constraint of an agent in sector $j$ is then given by

(1) \begin{equation} \sum _{k\in K^j(t)} p_k(t) c^j_k(t) + b^j(t) = a^j(t) + p_j(t)y^j(t) \end{equation}

and the demand for government bonds is assumed to be nonnegative:

(2) \begin{equation} b^j(t) \geq 0, \end{equation}

for all $j=1,\dots,N$ and $t\geq 0$ , that is, agents cannot borrow from the government.Footnote ⁶

When there are no private credit contracts, an equilibrium for the economy is characterized by a set of prices and allocations given a trajectory for fiscal and monetary policy such that each agent maximizes intertemporal utility and markets clear, that is

\begin{equation*} \sum _{j=1}^N c^j_k(t)=y^k(t) \end{equation*}

for all $t\geq 0$ . Market clearing in the goods markets implies that there is also market clearing for government bonds, such that

\begin{equation*} \sum _{j=1}^N b^j(t)=a(t) \end{equation*}

for all $t\geq 0$ , which in turn implies a government budget described by

\begin{equation*} a(t+1)=(1+i(t))a(t)-N\tau (t+1) \end{equation*}

that must hold in any period. We will append this economy with private credit contracts that will provide an additional tool of transferring resources between agents in a negative scenario.

2.2. An economy with trade credit contracts

We now consider the existence of a credit mechanism through which agents sign contracts in the beginning of the period to purchase goods from other agents and pay for the goods in the end of the period—trade credit. Trade credit agreements are an essential part of the economy, as firms without cash need to purchase inputs for production or supplies to be resold in advance.Footnote ⁷ In our model, trade credit is to be interpreted as a consequence of the scarcity of liquidity by the productive sector that needs financing to purchase goods that are essential to the productive process. For simplicity, we separate the entity that finances goods in each sector and assume that for each sector there exists a continuum of risk-neutral financial institutions that extend credit to consumers of the good produced by that sector before the realization of any uncertainty, collects payments from consumers after the realization of the shock and transfers net gains/losses to the representative consumer of the sector. These financial institutions are to be interpreted as the financial arm of the productive agents in the sector, and not as a separate financial sector. The amount of credit extended can only be used to purchase goods produced by that sector, so as to model a trade credit agreement. Agents may also purchase goods in the retail market after the realization of the shock, at the same price that they could buy the good through a trade credit contract, since prices are assumed to be predetermined before the occurrence of the shock. However, agents will pay an interest rate on credit contracts that will be determined by financial firms in a competitive scenario.

Since credit agreements are signed and payed in the same period, aggregate fluctuations are irrelevant to the mechanism, since they are known beforehand by all agents. For that reason, trade credit is irrelevant for periods $t \geq 1$ when there is no asymmetric imbalance. At period $t=0$ , however, agents may become insolvent due to a shock and not be able to pay for the contracts that were signed.

We assume that agents set the amount of credit demanded from suppliers by a fixed rule based on the consumption of the previous period. Given choices $c^j_k(t-1)$ that an agent in sector $j$ consumed at period $t-1$ of the good $k$ , this agent will then sign a contract in the beginning of period $t$ to purchase an amount $s^j_k(t)=s\big(c^j_k(t-1)\big)$ , where $s$ is a nonnegative function that specifies the amount of trade credit demand. This means that in period $t=0$ contracts $s^j_k(0)$ are based on an exogenous initial consumption level that we assume to be the optimal consumption that would be chosen by an altruistic social planner, in a normal scenario in which there is no asymmetric shock, that is, $c^j_k({-}1) = \alpha _{j-k} \bar{y}$ . We also assume that the rule that specify trade credit contracts is also rotationally symmetric, in the sense that the set $\left\{s^{j+1}_k(t)\right\}_{k=1}^N$ is equal to the rotation of $\left\{s^j_k(t)\right\}_{k=1}^N$ , so that there is no asymmetry that arises from the specification of trade credit contracts. This also implies that the total sum of credit contracts extended by each sector is the same for all sectors, that is, $\sum _k s^j_k(t) = \sum _k s^{j'}_k(t)$ , for all $j,j'=1,\ldots,N$ .

Since contracts were signed before the productivity shock was known of, some contracts may become unfeasible due to the lack of revenue from production and reduced demand. Additionally, previous trade credit contracts signed with a sector that is hit by the shock are nullified. This means that agents that were extended an amount $s^j_p(0)$ to consume on goods produced by the sector $p$ such that $\theta _p(0)=0$ are exempted from having to pay these contracts, since they cannot use these resources to spend on goods produced by sector $p$ .

Since contracts may become unfeasibile after the realization of an asymmetric shock, agents may need to default on a fraction of the amount they are required to pay, either by a bankruptcy law or renegotiation with creditors. Let $\psi ^j_k (\theta )$ be the fraction of default of an agent in sector $j$ with a contract signed with a supplier from sector $k$ , given a realization of the asymmetric shock $\theta$ .Footnote ⁸ Then, total profit of the financial firm in sector $j$ given a realization of the shock and default parameters is given by

\begin{equation*} \Pi ^j(\theta ) = \sum _{k=1}^N \left [\left(1-\psi ^k_j(\theta )\right) \left(1+r^j\right) -1 \right ] p_j(0) s^k_j(0) \end{equation*}

where $r^j$ is the interest rate charged by firms in sector $j$ . Firms are risk-neutral and set the real interest rate to maximize expected ex-ante profits, where the expectation refers to the possible occurrences of the asymmetric shock $\theta$ . Since there is a continuum of firms in competitive behavior, the equilibrium condition that determines the interest rate $r^j$ is simply $\mathbb{E} \Pi ^j (\theta ) = 0$ , which is solved by setting

(3) \begin{equation} 1+r^j = \frac{1}{1-\mathbb{E}\psi ^k_j(\theta ) \sigma ^k_j(0) } \end{equation}

where $\sigma ^k_j(0) = s^k_j(0)/\sum _k s^k_j(0)$ is the fraction of total credit extended to sector $k$ from sector $j$ . We assume that the default parameters $\psi ^j_k(\theta )$ also display a rotational symmetry with respect to the productivity shock $\theta$ , so that given a realization of the shock that hits sector $p$ , $\psi ^j_k(\theta )$ only depends on the relative position of $j$ and $k$ with respect to $p$ . This assumption is natural since all other aspects of the economy are symmetrical in the same way. This implies that the interest rates $r^j$ are equal for all sectors $j=1,\ldots,N$ .

An agent in sector $j$ in period $t=0$ chooses consumption levels $c^j_k(0)$ and bond holdings $b^j(0)$ , taking as given demand for goods $y^j(0)$ and profits from financial firms $\Pi ^j$ given an initial wealth $a^j(0)$ , subject to the constraint

(4) \begin{align} \sum _{k \in K^j(0)} p_k(0) c^j_k(0) & + \sum _{k \in K^j(0)} \left [ \left(1-\psi ^j_k(\theta )\right) \left(1+r^k\right) - 1 \right ] p_k(0) s^j_k(0) + b^j(0)\nonumber\\[5pt] & \leq a^j(0) + \Pi ^j (\theta ) + p_j(0) y^j(0) \end{align}

and subject to a lower-bound on consumption given the trade credit contract:

(5) \begin{equation} c^j_k(0) \geq s^j_k(0) \end{equation}

for all $k \in K^j(0)$ .

We are now in a condition to define the equilibrium outcome of the decentralized economy with trade credit.

It is easily seen that there can exist multiple choices for the parameters $\psi ^j_k(\theta )$ that are consistent with an equilibrium. For a trivial example, there is always an equilibrium in which all agents default on all trade credit agreements made, that is, $\psi ^j_k(\theta )=1$ for all $j=1,\dots,N$ , $k \in K^j(0)$ .

When there is no penalty to the default, instead of assuming the existence of trade credit as before, we can assume that defaults work as a transfer scheme after the productivity shock is learned of by the agents. This way of writing the problem is more convenient to determine the admissible parameters $\psi ^j_k(\theta )$ and allows us to use the analysis of transfers in Woodford (Reference Woodford2022) to study the effects of the bankruptcy law in this economy.

Rewrite the budget constraint (4) as

(6) \begin{equation} \sum _{k \in K^j(0)} p_k(0) c^j_k(0) + b^j(0) = \bar{a}^j(0) + p_j(0)y^j(0) \end{equation}

where

(7) \begin{equation} \bar{a}^j(0)=a^j(0)+\Pi ^j(\theta )-\sum _{k \in K^j(0)} \left [ \left(1-\psi ^j_k(\theta )\right) \big(1+r^k\big) - 1 \right ] p_k(0) s^j_k(0). \end{equation}

The initial nominal wealth $\bar{a}^j(0)$ is now composed of the fiscal transfers made by the government and the equivalent transfers that are made given the default parameters $\psi ^j_k(\theta )$ . In an equilibrium allocation with default, every agent in sector $j$ will maximize intertemporal utility subject to (6) for period $t=0$ , (1) for periods $t\geq 1$ , (2) for all periods with the additional constraint that $c^j_k(0)\geq s^j_k(0)$ for all $k \in K^j(0)$ since the credit good consumption was fixed before the beginning of the period.

We summarize this property in the following proposition:

Proposition 1. Assume initial trade credit quantities $s^j_k(0)$ . Then, an equilibrium with trade credit for an economy with policy trajectories $i(t)$ , $\tau (t+1)$ and $a^j(0)$ , default parameters $\psi ^j_k(\theta )$ for all $j$ and $k \in K^j(0)$ , trade credit interest rate $r^j$ that satisfy condition ((3)), and a productivity shock $\theta$ is an equilibrium to an economy without private credit contracts and with government policies $i(t)$ , $\tau (t+1)$ and $\bar{a}^j(0)$ , given by (7), with the added constraint:

(8) \begin{equation} c^j_k(0) \geq s^j_k(0) \end{equation}

for all $j=1,\ldots,N$ and $k \in K^j(0)$ .

The proposition shows that if condition (8) is met, defaults in credit contracts between sectors will cause part of the large drop in expenditure predicted by the network model in Woodford (Reference Woodford2022) to be accommodated by defaults on private credit contracts. The effect of these defaults can be thought of as transfers from agents in sectors that do not default to agents in sectors that do default. Additionally, agents in sectors in which the impact of the shock on consumption is larger are more likely to have a larger default parameter (in some cases, if initial wealth and fiscal transfers are small, the default parameter will necessarily be high), implying that the transfers are correctly targeted. This result has an important consequence for the optimal fiscal policy since the size of fiscal policy that will restore aggregate demand for the economy can be smaller.

As a simple example, consider the normal case such that $\theta = (1,\ldots,1)'$ and assume that $\psi ^j_k(\theta ) = 0$ for all sectors $j$ and goods $k$ . Then if predetermined prices $p_k(0)$ are the same for all goods $k$ , the rotational symmetry hypothesis for credit contracts and implies that $\bar{a}^j(0) = a^j(0)$ for all $j$ , that is, for every sector the profit of financial firms exactly equals the financial burden spent by the agents on trade credit contracts.

2.3. Optimal policy

In the “normal” case in which $\theta = (1,\ldots,1)'$ , we can assume that defaults are set to zero, and we have that $\bar{a}^j(0) = a^j(0)$ for all sectors. We can then analyze this economy as a simple decentralized economy without credit contracts, as in Woodford (Reference Woodford2022). In this case, we know that there is a solution to the decentralized economy with equal prices for all goods $p_j(t)=P(t)$ that admits the stationary solution $y(t)=\bar{y}$ , where $\bar{y}$ is such that $u'(\bar{y})=v'(\bar{y})$ and $c^j_k(t)=\alpha _{k-j}\bar{y}$ for all $t\geq 0$ , $j=1,\dots,N$ and $k\in K$ . A necessary condition is that $a^j(0)=a(0)/N$ so that each sector is endowed with the same initial nominal wealth. Another necessary condition for the optimal policy is the Euler condition

\begin{equation*} (1+i(t)) \frac {P(t)}{P(t+1)}=\frac {1}{\beta } \end{equation*}

to be satisfied for all $t \geq 0$ .Footnote ⁹ Additionally, the transversality condition

\begin{equation*} \lim _{t\to \infty }\beta ^t \frac {a(t)}{P(t)}=0 \end{equation*}

is required by the fiscal and monetary policy implemented by the government, where $a(t)=\sum _j a^j(t)$ . Finally, the government needs to implement a monetary policy that guarantees a stable path for the price level, such as a Taylor rule around a predetermined price level target trajectory $P^*(t)$ .Footnote ¹⁰

Equilibrium of the goods market implies that the government budget constraint is satisfied, with

\begin{equation*} a(t+1)=(1+i(t))a(t)-N\tau (t+1) \end{equation*}

for all $t \geq 0$ . We assume that fiscal policy is such that $a(t)\gt 0$ for all $t\geq 0$ . This condition implies that, in an equilibrium, there always exists a sector that does not have borrowing constraints. Clearing of the bond market requires that

\begin{equation*} \sum _{j=1}^N b^j(t)=a(t) \end{equation*}

for all $t\geq 0$ .

We assume that the government can, as a response to the occurrence of a productivity shock, make additional transfers to agents in period $t=0$ so that the quantity $a^j(0)$ on the budget constraint of agents represent both the initial wealth of agents and additional fiscal transfers (or taxes).

Assume that sector $p$ is the affected sector so that $\theta _p(0)=0$ . Woodford (Reference Woodford2022) shows that when the ex ante optimal policy $a^j(0)=a(0)/N$ is such that $a(0)\to 0$ , the expenditure equilibrium vector $\boldsymbol{c}=\big(c^1(0),\ldots,c^N(0)\big)'$ Footnote ¹¹ for this economy will be of the form $\theta \boldsymbol{\pi }$ , where $\boldsymbol{\pi }$ is the unique right eigenvector associated with the eigenvalue $1$ of the matrix $\textbf{A}$ , with $A_{jk}=\alpha _{j-k}/\big(1-\alpha _{p-k}\big)$ if $j\neq p$ , and $A_{pk}=0$ .Footnote ¹² It is then seen that a productivity shock to a single sector $p$ can cause major disruptions in the flow of payments, causing large reductions in the expenditure of multiple sectors, depending on the network structure that is specified.

It is also seen in Woodford (Reference Woodford2022) that a reduction in the interest rate $i(0)$ is not able to stimulate the affected sector $p$ and restore the first-best optimal equilibrium. Moreover, given any realization of the productivity shock $\theta$ , the optimal monetary policy can be shown to be leaving the nominal interest-rate unchanged, since stimulating the economy via an interest rate reduction makes the credit-constrained agents more constrained, while only raising demand from agents that are already on the optimal level of consumption. Therefore, we assume that monetary policy is unchanged when the shock is revealed, and we focus the analysis on fiscal transfers and the effect of credit default.

Now, in an economy without credit, if the fiscal policy is given by

\begin{align*} & a^{*p}(0)=\frac {a(0)}{N}+(1-\alpha _0)P(0)\bar {y} \\[5pt] & a^{*j}(0)=\frac {a(0)}{N}-\alpha _{p-j}P(0)\bar {y}\text {, for all $j \neq p$} \end{align*}

then it is shown that the first-best allocation is a transfer equilibrium of this economy.Footnote ¹³ We are interested in investigating whether the trade credit defaults described above are able to reduce the size of transfers that are required to be made since it may be unfeasible for a government to implement this fiscal policy during an adverse event. From here on, we always assume that the government follows an ex ante optimal policy so that $P(0)=\bar{p}$ is fixed with an equal price for every good, and the government follows an ex ante optimal monetary policy that stabilizes the economy in the optimal allocation when there is no shock.

2.4. A simple model with constant fraction of credit

We first illustrate the properties of the model in a simple example in an economy with a constant fraction of credit for each agent and assume that the default parameters are set exogenously.Footnote ¹⁴ We show the equilibrium outcome and calculate the reduction in the size of the optimal fiscal policy required to bring the economy to the first-best allocation for two network structures (chain and uniform networks) and different trade credit rules, such that $s^j_k(0)=\gamma ^j c^j_k({-}1)$ for all $k \in K^j(0)$ . This setup implies that $s^j_k(0)=\gamma ^j\alpha _k\bar{y}$ for every $j=1,\ldots N$ , $k \in K^j(0)$ . Assume in both cases that $\bar{p}$ is the equilibrium price for period $t=0$ when there is no shock and initial fiscal policy $a^j(0)=a(0)/N$ is small. We denote by $p$ the sector such that $\theta _p(0)=0$ when the economy is hit by a negative productivity shock.

Now, both the distribution of the credit parameters $\gamma ^j$ and the network structure specified will be critical to determine whether there exists default parameters $\{\psi ^j_k(\theta )\}_{j=1,..N\text{, }k \in K^j(0)}$ that can be implemented by a bankruptcy law to reduce the size of fiscal transfers that restore the optimal allocation. We explore two cases: in the first case, each sector has the same quantity of credit, $\gamma ^j=\gamma$ , while in the second case, a single sector $j\neq p-1$ does not purchase any goods through credit, that is, $\gamma ^j=0$ . In the first case, it is easily seen that there always exists a set of default parameters that reduces the total size of fiscal transfers, exactly by a fraction of $\gamma$ , regardless of the network structure. In the second case, however, the network structure will be critical for the bankruptcy law to allow for any reduction on fiscal policy.

In the first example, we assume that all sectors have the same amount of consumption financed through trade credit so that $\gamma ^j=\gamma$ for every $j=1,\ldots N$ . In this case, the network structure is irrelevant to determine the size of the reduction in the optimal fiscal policy.Footnote ¹⁵

Computing the equivalent transfers $\bar{a}^j(0)$ given arbitrary default parameters, we obtain

\begin{equation*} \bar {a}^j(0)=\frac {a(0)}{N}+\gamma \bar {p}\bar {y} \left \{\sum _{k=1}^N\left[ \left(1-\psi ^k_j\right)(1+r)-1\right]\alpha _{k-j} - \sum _{k \in K^j(0)}\left[ \left(1-\psi ^j_k\right)(1+r)-1\right]\alpha _{j-k}\right \} \end{equation*}

for $j \neq p$ , while

\begin{equation*} \bar {a}^p(0)=\frac {a(0)}{N}-\gamma \bar {p}\bar {y}\sum _{k \in K^p(0)} \left[\left(1-\psi ^p_k\right)(1+r)-1\right]\alpha _{p-k} \end{equation*}

where $r$ is the common interest rate charged by all sector on trade credit contracts. Then, it is easily seen by setting $\psi ^j_k=1$ for every $j=1,..N$ , $k \in K^j(0)$ , that the equivalent transfers can be set as

\begin{equation*} \bar {a}^j(0)=a^j(0)-\gamma \bar {p}\bar {y} \alpha _{p-j} \end{equation*}

for $j \neq p$ , and

\begin{equation*} \bar {a}^p(0)=a^j(0)+\gamma \bar {p}\bar {y} (1-\alpha _0). \end{equation*}

If the government then implements a reduced fiscal policy of

(9) \begin{equation} a^j(0)=\frac{a(0)}{N}-(1-\gamma ) \bar{p}\bar{y} \alpha _{p-j} \end{equation}

for $j \neq p$ , and

(10) \begin{equation} a^p(0)=\frac{a(0)}{N}+(1-\gamma ) \bar{p}\bar{y} (1-\alpha _0) \end{equation}

then we obtain the equality $\bar{a}^j(0)=a^{*j}(0)$ for all $j=1,\ldots,N$ and the first-best allocation is restored. In this case, a permissive bankruptcy law that allows all agents to default on their contracts temporarily allows for a reduction of the necessary size of the fiscal policy that restores the optimal allocation. However, setting $\psi ^j_k=1$ for every $j=1,..N$ and $k \in K^j(0)$ is not the only possible choice of parameters that allows this same reduced fiscal policy to be implemented. Setting $\psi ^j_j=0$ for every $j$ for instance is immaterial. Additionally, different network structures allow different default parameters to be implemented.

• • In the chain network structure with parameter $\lambda$ , we necessarily have to set $\psi ^j_k=1$ so that each sector defaults on all credit taken with the adjacent sector, with the exception of sector $p-1$ . Fig. 2 illustrates transfers made through the bankruptcy law for $N=4$ :

• • In the uniform network structure, the same policy can be implemented simply by setting $\psi ^p_k=1$ for every $k \neq p$ , while all other parameters can be set to any level, even to zero. This means that in the uniform network, only sector $p$ goes bankrupt and defaults. All other sectors are still solvent and can honor the credit agreements without defaulting.

Figure 2. Bankruptcy in the chain network case with constant fraction of credit. Arrows represent the size of default from each sector. The net effect is a transfer from sector 4 to sector $p=1$ .

We summarize this result with a proposition:

In a second formulation, assume now that there is a single sector $l$ such that $\gamma ^l=0$ , that is, agents in this sector do not finance their consumption through trade credit, while they extend credit to their customers. In this case, the specified network structure now deeply influences the attainable transfers through credit defaults. To illustrate this idea, we study the two networks considered previously: the chain network and the uniform network.

In the chain network structure, assume that $l\neq p-1$ . Computing the equivalent transfers to sector $l$ we obtain

\begin{equation*} \bar {a}^l(0)=a^l(0)-\psi ^{l-1}_l \gamma ^{l-1} \lambda \bar {p}\bar {y} \end{equation*}

which implies that if $\psi ^{l-1}_l\gt 0$ , the only way that this sector’s demand can be restored is if the government transfers additional resources to sector $l$ to outweigh the reduction caused by the sector $l-1$ default. Since in the optimal transfer policy, the only sector that should be taxed is sector $p-1$ , we see that a bankruptcy law in this case will be ineffective to restore the optimal allocation. We illustrate in Fig. 3 this case with $N=4$ and all other default parameters $\psi ^j_{j+1}=1$ :

Figure 3. In the chain network, when a single sector does not finance consumption by credit, transfers implemented by the bankruptcy procedure may not be correctly targeted.

In the uniform network case, there is a different outcome to this situation. As we have seen before, we can always implement a fraction of the optimal fiscal policy through credit defaults in the uniform case by setting sector $p$ parameters $\psi ^p_k=1$ and all other parameters to zero. In this case, the only credit parameter of importance is $\gamma ^p$ , and setting the credit parameter $\gamma ^l=0$ of any sector $l\neq p$ is inconsequential to this result. The connectedness of the uniform network implies that it is easier to implement an adequate bankruptcy law that transfers resources to the affected sector.

3.1. Credit markets and autarky

In the general setup as in Section 2, assume that for every sector $j$ we can decompose the sets $K^j(0)$ in two sets $\mathcal{S}^j$ and $\mathcal{R}^j=K^j(0)\backslash \mathcal{S}^j$ . The set $\mathcal{S}^j$ denotes the supplier sectors of sector $j$ so that all consumption from these sectors must be credit-financed. The set $\mathcal{R}^j$ are the retailers,Footnote ¹⁶ and all consumption from these sectors are cash-financed. We do not assume a priori that the sets of suppliers and retailers are symmetric.

As before, given a policy trajectory from the government, each agent plans consumption and asset accumulation and makes trade credit contracts $s^j_k(t)=c^j_k(t-1)$ for $k \in \mathcal{S}^j$ , with $s^j_k(0)=c^j_k({-}1)$ given by the ex ante optimal consumption, and $s^j_k(t)=0$ for $k \in \mathcal{R}^j$ . An agent can choose to default on credit agreements made before regardless of the occurrence of a productivity shock, but this agent is excluded from purchasing from supply markets ever again.Footnote ¹⁷ For an equilibrium to exist, we also assume that if a sector $j$ is such that $\mathcal{R}^j=\{j\}$ , then in autarky, other sectors $k\neq j$ are also excluded from purchasing goods from sector $j$ .Footnote ¹⁸ The optimization problem for the agent in sector $j$ in autarky for periods $t\geq 1$ is maximizing

\begin{equation*} \sum _{t=0}^\infty \beta ^t U^{j,aut}(t) \end{equation*}

where

\begin{equation*} U^{j,aut}(t)=\sum _{k \in \mathcal {R}^j}\theta _k(0)\alpha _{k-j} u\left (\frac {c^j_k(t)}{\alpha _{k-j}\theta _k(0)}\right ) \end{equation*}

for $t\geq 1$ , with the usual utility function for period $t=0$ with $c^j_k(0)=s^j_k(0)$ for $k \in \mathcal{S}^j$ , and an agent in autarky in period $t=0$ is subject to the budget constraint

\begin{equation*} \sum _{k \in \mathcal {R}^j}p_k(0)c^j_k(0)+b^j(0)=a^j(0)+\Pi ^j+p_j(0)\bar {y} \end{equation*}

and

\begin{equation*} \sum _{k \in \mathcal {R}^j}p_k(t)c^j_k(t)+b^j(t)=a^j(t)+p_j(t)\bar {y} \end{equation*}

for periods $t\geq 1$ .

The individual rationality condition for agent $j$ not to choose to default when there is no negative shock is simply that

\begin{equation*} \sum _{t=0}^\infty \beta ^tU^j(t) \geq \sum _{t=0}^\infty \beta ^tU^{j,aut}(t). \end{equation*}

It is straightforward to obtain the following proposition:Footnote ¹⁹

This proposition states that when there is no adverse shock, agents will not default on the credit taken with suppliers, for a value of $\beta$ close enough to $1$ .Footnote ²⁰ This means that the penalty of excluding agents from markets is sufficient, in this model specification, to guarantee that credit markets function as in a complete market model, and the penalty is never actually enforced. However, when there is a productivity shock, agents in affected sectors will have no alternative but to default.

3.2. Optimal bankruptcy law

We assume that the government follows an optimal initial policy such that $a(0)\to 0$ before the productivity shock is known of. As a consequence, agents hit by an adverse shock will have no alternative but to default and be penalized by being excluded from credit markets. This situation will be suboptimal both for the sector that defaults and its creditors, in the case of the two network structures that were presented above.

In an economy with a chain network structure with parameter $\lambda$ , assume that $\mathcal{S}^j=\{j+1\}$ and $\mathcal{R}^j=\{j\}$ . The optimal initial policy guarantees that if there is no default in the economy, each agent consumes $\lambda \bar{y}$ of the good produced by the supplier and $(1-\lambda )\bar{y}$ of the own sector’s goods.

In this case, when the economy is hit by a negative productivity shock to a sector $p$ , then this sector defaults in period $t=0$ , and the only possible equilibrium is for every sector to also default, except for sector $p-1$ , since it will not have sufficient funds to cover its debt.Footnote ²¹ Since all sectors default, they are penalized with autarky so that for $t\geq 1$ each sector $j=1,\ldots,N$ will consume only its own goods. The utility from this autarky in any period $t\geq 1$ is at most $(1-\lambda )u(\bar{y}/(1-\lambda ))$ which is strictly dominated by the utility of $u(\bar{y})$ that is attained when there is no default, by the strict concavity of the utility function.

Therefore, a bankruptcy law that pardons the trade credit debt in period $t=0$ when there is a negative shock will be optimal not only for the affected sector but also for all agents, including creditors. By the equivalence proposition of the previous subsection, the full pardon of the debt of size $\lambda \bar{p}\bar{y}$ in period $t=0$ will be equivalent to a fiscal transfer that is financed by sector $p-1$ , with multiplier $\lambda ^{-1}$ to sector $p+1$ .Footnote ²² Sector $p+1$ will then spend $\bar{p}\bar{y}$ in period $t=0$ , and so will sector $p+2$ and so on, so that every sector of the economy is able to consume exactly the first-best optimal allocation of this economy. In this case, no additional fiscal policy is needed to support the first-best equilibrium.

In an uniform network structure, regardless of the sets $\mathcal{R}^j$ and $\mathcal{S}^j$ specified, when sector $p$ is hit by a productivity shock all agents of this sector necessarily default as in the chain network case, but the budget constraint of agents in other sectors still allows them not to default, as long as the supplier and retailer specifications $\mathcal{R}^j$ and $\mathcal{S}^j$ are the same for all sectors $j=1,\ldots,N$ . In this case, only sector $p$ will be in autarky in periods $t\geq 1$ .

In autarky, agents in sector $p$ can only purchase goods from sectors $k \in \mathcal{R}^p$ . It can be shownFootnote ²³ that any equilibrium in this scenario is such that there are two stationary relative prices $q_S \lt 1 \lt q_R$ , where $q_k=p_k(t)/P(t)$ and $q_k=q_R$ if $k \in \mathcal{R}^p$ and $q_k=q_S$ otherwise. Reduced demand from sector $p$ implies that $p$ -supplier sectors will have a smaller equilibrium relative price from $t=1$ forward. This implies that they will purchase a smaller total quantity of goods in equilibrium since they will still desire to consume goods from $p$ -retailer sectors that are now relatively more expensive. This means that intertemporal utility of $p$ -suppliers is smaller when sector $p$ is in autarky, and it will be optimal for these agents to lift the consequences of the default.

The previous examples show that lifting consequences applied to agents that default during a productivity shock constitutes not only an alternative to a targeted fiscal policy to sectors affected but also improves the welfare of their creditors. This happens because the usual bankruptcy procedure that excludes agents that default from markets, while being effective in preventing agents from deliberately choosing to default in normal times, also punishes creditors by reducing their access to goods produced by defaulters.

An important advantage of this bankruptcy procedure is that it only requires the government to understand which sectors are directly affected by the shock, so that their penalties are lifted ex-post. Since these sectors will default regardless of the existence of penalties, the affected sectors will be exactly those that default in the trade credit contracts. Solvent sectors will not wish to default, as they will expect to suffer the exclusion from credit markets as a consequence.Footnote ²⁴ For this reason, bankruptcy policies can aid policymakers in countries subject to extreme unexpected events, by alleviating the consequences of these shocks in those most affected sectors, while requiring less information of the structure of the economy from the government than direct fiscal transfers.

We conclude this subsection with a general proposition:

Proposition 4. Let an economy with trade credit be characterized by a symmetric credit structure so that $\mathcal{S}^{j+1}=R(\mathcal{S}^j)$ for every $j=1,\ldots,N$ , where $R$ is the rotation operation. Given a productivity shock to sector $p$ , an ex ante optimal fiscal and monetary policy and initial constant price $\bar{p}$ , there always exists an exogenous bankruptcy law that guarantees that the modified initial nominal wealth is given by

\begin{equation*} \bar {a}^p(0)=a^p(0)+\alpha _{S^p}\bar {p}\bar {y} \end{equation*}

where $\alpha _{S^p}=\sum _{k \in S^p}\alpha _k$ and

\begin{equation*} \bar {a}^j(0)=a^j(0)-\alpha _{p-j}\bar {p}\bar {y} \end{equation*}

for every $j$ that is a $p$ -supplier.

In addition, a fiscal policy given by

\begin{equation*} a^p(0)=\frac {a(0)}{N}+(1-\alpha _{S^p}-\alpha _0)\bar {p}\bar {y} \end{equation*}

and

\begin{equation*} a^j(0)=\frac {a(0)}{N}-\alpha _{p-j} \bar {p}\bar {y} \end{equation*}

for every $j\neq p$ that is a $p$ -retailer, restores the optimal allocation. In particular, if $R^p=\{p\}$ , no additional fiscal policy is needed.

This proposition is straightforward from the definition of the modified initial nominal wealth $\bar{a}^j(0)$ by setting $\psi ^j_k=1$ for every $j=1,\ldots,N$ and $k \in S^j$ . However, this choice of default parameters is not unique, and other choices of default parameters can be implemented through the bankruptcy law to obtain the transfers described in the proposition.

3.3. Credit renegotiation

We now extend the example with penalties to the debtor by autarky previously analyzed and consider the case in which creditors accept to renegotiate debt from agents that are insolvent due to the adverse shock. Given initial credit contracts $s^j_k(0)$ and a productivity shock to sector $p$ , creditors and insolvent debtors agree to an alternative payment scheme in future periods $t\geq 1$ so that creditors are compensated and debtors are allowed to default in period $t=0$ and postpone the payment of credit contracts.

Assume that agents either pay or default in full their credit commitments. Denote by $D \subset \{1,\ldots,N\}$ the subset of sectors that default in period $t=0$ . An agent that defaults is punished by autarky, which is undesirable both to the debtor and the creditor. Assume that agents negotiate a delayed payment scheme, represented by a sequence $\{h^j(t)\}_{t\geq 1}$ of nonnegative amounts that extends to infinity, satisfying

\begin{equation*} \sum _{t \geq 1} Q_{1,t} h^j(t) = (1+i(0))\sum _{k \in S^{j}}p_k(0)s^j_k(0) \end{equation*}

where $Q_{1,t}=\prod ^{t-1}_{s=1}\frac{1}{1+i(s)}$ and $Q_{1,1}=1$ is the discount factor for future cash flows. In this sense, the creditor is paid out in full as the present value of future payments equals the amount that was agreed to be paid at period $t=0$ .

Assuming that the government follows an ex ante optimal policy so that initial prices $\bar{p}$ are equal, the sequence $a(t)$ is also assumed to satisfy a condition that guarantees that the borrowing constraint never binds.Footnote ²⁵ In this case, the existence of equilibrium is guaranteed, with a default parameter of $\psi ^j=1$ for all $j\in D$ , and a different borrowing constraint that consider the existence of future period payments:

\begin{equation*} \sum _{k \in K^j(0)} p_k(t)c^j_k(t)+b^j(t)+\unicode {x1D7D9}_{j\in D}h^j(t) = a^j(t)+\sum _{k \in D} h^k(t) + p_j(t)y^j(t). \end{equation*}

where $\unicode{x1D7D9}_{j\in D}=1$ if $j \in D$ and $0$ otherwise. It can then be shownFootnote ²⁶ that if we assume as in Woodford (Reference Woodford2022) that $\beta \to 1$ , then the existence of future payments will not disrupt the equilibrium efficient allocation, as these payments will become more insignificant to determine consumption choices as the agent becomes more patient. We can then conclude that a mutually beneficial alternative of credit renegotiation can be agreed to by creditors and debtors that allows agents in period $t=0$ to consume in this period without generating a penalty for defaulters. This more laissez-faire alternative is equivalent to a government policy that taxes creditors in period $t=0$ to transfer to debtors and then taxes debtors in future periods to compensate creditors.

To compare with other bankruptcy law alternatives, assume that the government also implements a residual fiscal policy in period $t=0$ that is optimal, in the sense that it taxes cash consumers of the affected sector $p$ and transfers to sector $p$ , as dictated by the optimal policy rule. In this case, if insolvent sectors in period $t=0$ are allowed to default on credit contracts, then demand will be restored to the first-best level considering a negative shock. There are then three cases: (i) if sector $p$ defaults and is punished by autarky, there is a suboptimal allocation in all periods $t\geq 1$ for all sectors, including creditors; (ii) if a bankruptcy law allows for insolvent sectors to default freely, then the first-best level is restored for all periods; and (iii) if there is an endogenous credit renegotiation that establishes a future compensation for creditors that suffer default, then there is a distortion of future allocations in favor of creditors that vanish as $\beta \to 1$ .

A different type of credit policy is considered in Woodford (Reference Woodford2022), in which the government is able to relax the borrowing constraint of different sectors, allowing agents to borrow from the government during the occurrence of a negative shock. While this policy can stimulate those sectors most affected by the shock, it cannot replicate the optimal allocation, with the exception of the limiting case when $\beta \to 1$ , since it distorts future consumption when agents need to pay for the credit. While similar to the credit renegotiation scenario analyzed in this section, the relaxation of borrowing constraints requires full information by the government of the sectors that need to be stimulated. An advantage of bankruptcy procedures such as the trade credit defaults postulated in this paper is that they depend less on the information available by the government, since agents themselves will choose to default or renegotiate credit contracts. The role of the government is to temporarily allow for this mechanisms to function by changing the prevailing bankruptcy law and provide incentives to credit renegotiation.