Breaking bad: supply chain disruptions in a streamlined agent-based model

Abstract

We explore the macro-financial consequences of the disruption of a supply chain in an agent-based framework characterized by two networks, a credit network connecting banks and firms and a production network connecting upstream and downstream firms. We consider two scenarios. In the first one, because of the lockdown, all the upstream firms are forced to cut production. This generates a sizable downturn during the lockdown due to the indirect effects of the shock (network-based financial accelerator). In the second scenario, only those upstream firms located in the ‘red zone’ are forced to contract production. In this case, the recession is milder and the recovery begins earlier. Upstream firms hit by the shock, in fact, will be abandoned by their customers who will switch to suppliers who are located outside the red zone. In this way, firms endogenously reconstruct (at least in part) the supply chain after the disruption. This is the main determinant of the mitigated impact of the shock in the ‘red zone’ type of lockdown.

Keywords:

• Supply chain
• credit network
• financial accelerator
• lockdown

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 See, for instance, Assenza, Delli Gatti, and Grazzini (2015) and Assenza et al. (2018). For an exhaustive survey of macroeconomic agent-based model, see Dawid and Delli Gatti (2018).

2 See, for instance, Dolgui, Ivanov, and Sokolov (2018).

3 For an insightful exhaustive overview of this literature, see Antras (2016) and the references therein.

4 See Battiston et al. (2007, 2012) for previous work on the production/trade credit network among firms and for contagion in financial networks. Riccetti, Russo, and Gallegati (2013) have enriched the framework put forward by Delli Gatti et al. (2010) with a more sophisticated theory of leverage determination.

5 The ex post marginal cost for each D-firm will be based on the actual price of intermediate inputs the firm will pay, which has a firm-specific component. See Section 3.4.

6 The opportunity cost of internal finance is the risk free interest rate r which will be introduced below, see Section 3.5.

7 If the marginal benefit of screening were linearly increasing with size, e.g. $z=\lambda A_i$, the first-order condition would read $L_i=\lambda A_i$. This rule can be interpreted as follows: the bank sets the optimal size of the loan by applying a leverage target λ to the firm's net worth. In this case, $F_i=(1+\lambda )A_i$. Substituting this expression in (8(8) $Y_i=\frac{F_i}{c_D}=(1+\mu )F_i.$(8) ), we get the FY function: $Y_i=(1+\mu )(1+\lambda )A_i$. In this case, the FY function of the borrowing firm would be linear.

8 An alternative simple microfoundation of the concave FY function can be found following the approach pioneered by Greenwald and Stiglitz (1993). The problem of the firm consists in maximizing expected profits net of bankruptcy costs. Suppose that there are two states of the world. In the positive or favorable state – which occurs with probability $(1-{\varphi }_i^b)$ – the firm earns operating profits ${\pi }_i^s=(\mu /(1+\mu ))Y_i$. In the negative or unfavorable state of the world – which occurs with probability ${\varphi }_i^b$ – the firm goes bankrupt and earns profits ${\pi }_i^b={\pi }_i^s-C_i^b$ where $C_i^b=(1/2)Y_i^2$ is the cost of bankruptcy. The firm maximizes the expected value of profits: $\underset{Y_i}{max}{V}_i={\pi }_i^s-C_i^b{\varphi }_i^b=\frac{\mu }{1+\mu }{Y}_i-\frac{{\varphi }_i^b}{2}{Y}_i^2$The (closed form) solution to this problem is $Y_i=\mu /((1+\mu ){\varphi }_i^b)$. Let us assume now that the probability of bankruptcy is a decreasing (convex) function of financial robustness captured by net worth: ${\varphi }_i^b=A_i^{-\beta };A_i\ge 1$. Substituting this expression into the solution above, we get the FY function $Y_i=(\mu /(1+\mu ))A_i^{\beta }$. Also in this setting, the FY function is increasing and concave.

9 The prices in question are real prices, i.e. ratios of the corresponding nominal price of intermediate inputs to the GDP deflator.

10 In real world contracts, if the customer pays on delivery, she will get a discount: $p_{j,c}^i=d_j^i{p}_j^i$ with $0<d_j^i<1$. It is straightforward to interpret the discount as the reciprocal of the gross interest rate on trade credit.

11 We assume that $u_j$ has a normal distribution with zero mean and standard deviation ${\sigma }_u=0.3$: $u_j\sim \mathcal{N}(0,0.3)$. We set the standard deviation such that the realizations of the random variable fall in the interval $-0.9<u_j<0.9$ with probability 0.997. This assumption assures the non-negativity of the marginal cost $c_j$ and therefore of the price $p_{j,c}^i$.

12 The fundamental component of the cash price of intermediate inputs has been already introduced above. It contributes to the determination of the ex ante marginal cost for D-firms (see Assumption 3.2).

13 The relationship could even be ‘downward sloping’. Consider, for instance, two U-firms, say U1 and U2, with $A1\ll A2$. Suppose the relatively ‘poor’ U1 has many and/or rich D-customers such that $Q_1$ is ‘high’ while the relatively rich U2 has few and/or poor D-customers so that $Q_2$ is ‘low’. Hence, it could be the case that $Q_1>Q_2$.

14 In words: firms indexed with $f\in [1,N_D]$ produce D-goods; firms indexed with $f\in [N_D+1,N_F]$ produce intermediate goods.

15 We adopt the wording of the literature on financial frictions. For the pioneering work, see Bernanke, Gertler, and Gilchrist (1999).

16 Notice that, if the old and the new partners were both ‘wealthy’, i.e. if they did not have to collect funds in the form of deposits, the leverage of each bank would be zero so that the firm would be unable to make a decision. In this case, in simulating the model, we assume that the firm will decide to switch to the new partner if the latter's net worth is bigger than the net worth of the old partner.

17 The crucial variable $R_{0,1}^j$ takes the form of a ‘relative price’ and has two components, a deterministic time varying component (relative interest rate) which is characterized by the leverage of the borrower and the lender (new and old) and a stochastic component, i.e. the ratio of the idiosyncratic shocks of the post-shipment price. In running simulations, we realized that, since interest rates have a low variability, the stochastic component overrides the relative interest rates and plays a major role in shaping the production network. Therefore, in this formulation, the allocation of links to nodes is distributed in an almost completely random way, with very little influence of financial considerations on partner choice. In our view, this makes the analysis not particularly interesting. Therefore, in the simulations we have used a simplified definition of the crucial variable, namely: $R_{0,1}^j=\frac{{\lambda }_{j_0}^{\rho }+({\lambda }_i^U{)}^{\rho }}{{\lambda }_{j_1}^{\rho }+({\lambda }_i^U{)}^{\rho }}.$In other words, we assume that the D-firm will choose the U-firm which is in a better shape from the financial point of view, abstracting from the short run oscillations of the price. Financial robustness is proxied by leverage.

18 Notice that if the old and the new U-partners were both self-financed, i.e. if they had zero leverage – the D-firm would be unable to make a decision. In this case, in simulating the model, we assume that the D-firm will decide to switch to the new U-partner if the latter's net worth is bigger than the net worth of the old U-partner.

19 From simulations, we inferred that a smaller average size at entry would lead to an unrealistic wave of early defaults of newly born U-suppliers.

20 For each variable, (i) we apply the HP filter to the time series generated by each simulation, (ii) we compute the average of the filtered time series and (iii) we use the averaged time series to compute the long run mean and standard deviation.

21 The correlation indexes are computed on the same filtered and averaged data used for Table 2.

22 We decided to take a picture of the connections in the production network in period t = 449 of the baseline and of the red zone scenario. This is the last period of the localized lockdown (which goes from period 400 to period 450) in the red zone scenario.

Additional information

Notes on contributors

Domenico Delli Gatti

Domenico Delli Gatti is Professor in Economics at Università Cattolica del Sacro Cuore in Milan. He is also Director of the Complexity Lab in Economics, a research center affiliated to the Department of Economics and Finance of the same University. His primary research interests are macroeconomics with heterogeneous interacting agents (agent based macroeconomics), business cycles and monetary economics.

Elisa Grugni

Elisa Grugni is PhD student in the PhD program in Economics and Finance at Università Cattolica del Sacro Cuore in Milan. Her primary research interest is macroeconomics with particular reference to models with heterogeneous interacting agents.