The co-evolution of integrated corporate financial networks and supply chain networks with insolvency risk

Abstract

In this paper, we develop a modeling framework that integrates supply chain networks and corporate financial networks, and studies the co-evolution of the two types of networks. The framework allows one to investigate the following questions: (1) How do the physical product flows, demands and prices interplay with the corporate financial networks, and evolve over time with varying financial and economic conditions? (2) How do the financial flows, profits, and insolvency risks interplay with the supply chain networks, and evolve over time with varying financial and economic conditions? We illustrate the modeling framework with computational studies. In particular, the computational results demonstrate how the two types of networks interplay and evolve in a business environment similar to the one during the 2008–2009 financial crisis in which the credit markets tightened up and the product demands continued to decline.

Appendix

1.1 Qualitative properties

We now we provide conditions for existence of a solution to variational inequality (10). We also prove that the $F(X)$ given by (10) that enters the variational inequality (1) is monotone.

Theorem 2

Existence If $(Q^{1t*}, Q^{2t*}, Q^{3t*}, \lambda ^{t*}, \gamma ^{t*})$ satisfies variational inequality (10) then $( Q^{1t*}, Q^{2t*}, Q^{3t*})$ is a solution to the variation inequality problem: Determine $( Q^{1t*}, Q^{2t*}, Q^{3t*} )\in \mathcal{K }^1$ satisfying:

$$\begin{aligned}&\sum _{i=1}^I\sum _{j=1}^J \left[{\partial h_{ij1}^t(q_{j1}^{it*},\Pi _{t-1}^i)\over \partial q_{j1}^{it}}+ c_{it}+{\partial \hat{h}_{ij1}^t(q_{j1}^{it*}, \Pi _{t-1}^j)\over \partial q_{j1}^{it}}\right]\times \left[ q_{j1}^{it}-q_{j1}^{it*}\right]\nonumber \\&\quad +\sum _{i=1}^I\sum _{j=1}^J \left[{\partial h_{ij2}^t(q_{j2}^{it*},\Pi _{t-1}^i)\over \partial q_{j2}^{it}}+{\partial r_{ij}^t(q_{j2}^{it*},\Pi _{t-1}^j)\over \partial q_{j2}^{it}}+ c_{it}+{\partial \hat{h}_{ij2}^t(q_{j2}^{it*}, \Pi _{t-1}^j)\over \partial q_{j2}^{it}}\right]\nonumber \\&\quad \times \left[ q_{j2}^{it}-q_{j2}^{it*}\right]+\sum _{j=1}^J \sum _{m=1}^M\left[ c_{jt}-\rho _m^{t}\left(\sum _{j=1}^Jq^{jt*}_m\right)- {\partial \rho _m^{t}\left(\sum _{j=1}^Jq^{jt*}_m\right)\over \partial q^{jt}_m}q^{jt*}_m \right]\nonumber \\&\quad \times \left[ q^{jt}_m-q^{jt*}_m\right] \ge 0,\quad \forall (Q^{1t},Q^{2t},Q^{3t})\in \mathcal{K}^1, \end{aligned}$$

(31)

where $\mathcal{K }^1\equiv ((Q^{1t},Q^{2t},Q^{3t})|(Q^{1t},Q^{2t},Q^{3t})\in R_+^{2IJ+JM}$ and (3) and (7) hold).

A solution to (31) is guaranteed to exist provided that the cost functions, the risk functions, and the inverse demand functions are continuously differentiable and the feasible set is nonempty. Moreover, if $( Q^{1t*}, Q^{2t*}, Q^{3t*} )$ is a solution to (31) then there exist $(\lambda ^{t*}, \gamma ^{t*})$ with $(Q^{1t*}, Q^{2t*}, Q^{3t*},\lambda ^{t*}, \gamma ^{t*})$ being a solution to variational inequality (10).

Proof

Note that the feasible set of (31) is convex, and bounded by constraints (3) and (7). So, the feasible set of (31) is convex and compact. The existence of a solution to (31) is guaranteed from the standard theory of variational inequality since the cost functions, the risk functions, and the inverse demand functions are continuously differentiable, and the feasible set is convex, compact, and nonempty (cf. Nagurney (1993)). The rest of the proof is an analog to Theorem 3 in Nagurney and Dhanda (2000).$\square$

Theorem 3

Monotonicity Suppose that all cost functions and risk functions in the model are continuously differentiable and convex, and the inverse demand functions are continuously differentiable and monotonically decreasing. Then the vector F that enters the variational inequality (10) as expressed in (1) is monotone, that is,

$$\begin{aligned} \left<(F(X^{\prime })-F(X^{\prime \prime }))^T,X^{\prime }-X^{\prime \prime }\right> \ge 0,\quad \forall X^{\prime },X^{\prime \prime }\in \mathcal{K }, X^{\prime }\ne X^{\prime \prime }. \end{aligned}$$

(32)

Proof

We can verify that if all cost functions and risk functions in the model are continuously differentiable and convex, and the inverse demand functions are continuously differentiable and monotonically decreasing, then the vector F that enters the variational inequality is convex. Therefore, the vector F is monotone.$\square$

1.2 The modified projection method

In the computational procedures, variational inequality (10) as expressed in (1) is solved using the modified projection method [see Korpelevich (1977) and Nagurney (1993)]. The method converges to a solution of the model provided that $F(X)$ is monotone and Lipschitz continuous, and a solution exists, which is the case of the our model.

• Step 0 Initialization Start with $X^0\in \mathcal{K }$ where $\mathcal{K }\equiv R^{2IJ+IM+I+J}_+$, and select $\alpha$, such that $0<\alpha \le {1\over L}$, where $L$ is the Lipschitz constant for function $F(X)$. Let $n:=1$.

• Step 1 Construction and computation Compute ${\bar{X}}^{n-1}$ by solving the variational inequality subproblem:

  $$\begin{aligned} \left\langle ({\bar{X}}^{n-1}+(\alpha F(X^{n-1})-X^{n-1}))^T,X-{\bar{X}}^{n-1}\right\rangle \ge 0,\quad \forall X\in \mathcal{K}. \end{aligned}$$

  (33)

  In particular, the explicit formulae for the solution to variational inequality (34) are as follows:

  $$\begin{aligned} {\bar{q}}^{itn-1}_{j1}&= \text{ max}\left( 0,q^{itn-1}_{j1}+\alpha \left( \theta _{j}\gamma _{j}^{tn-1}-{\partial h_{ij1}^t(q_{j1}^{itn-1},\Pi _{t-1}^i)\over \partial q_{j1}^{it}}- c_{it}-\lambda ^{itn-1}\right.\right.\nonumber \\&\qquad \qquad -\left.\left.{\partial \hat{h}_{ij1}^t(q_{j1}^{itn-1}, \Pi _{t-1}^j)\over \partial q_{j1}^{it}}\right)\right),\end{aligned}$$

  (34)
  $$\begin{aligned} {\bar{q}}^{itn-1}_{j2}&= \text{ max}\left(0,q^{itn-1}_{j2}+\alpha \left(\theta _{j}\gamma _{j}^{tn-1}-{\partial h_{ij2}^t(q_{j2}^{itn-1},\Pi _{t-1}^i)\over \partial q_{j2}^{it}}-{\partial r_{ij}^t(q_{j2}^{itn-1},\Pi _{t-1}^j)\over \partial q_{j2}^{it}}\right.\right.\nonumber \\&\qquad \qquad -\left.\left.c_{it}-\lambda ^{itn-1} -{\partial \hat{h}_{ij2}^t(q_{j2}^{itn-1}, \Pi _{t-1}^j)\over \partial q_{j2}^{it}}\right)\right),\end{aligned}$$

  (35)
  $$\begin{aligned} {\bar{\lambda }}^{itn-1}&= \text{ max}\left(0,\lambda ^{itn-1}+\alpha \left(\sum _{j=1}^J\left(q_{j1}^{itn-1}+q_{j2}^{itn-1}-PCAP_{it}\right)\right)\right),\end{aligned}$$

  (36)
  $$\begin{aligned} {\bar{q}}^{jtn-1}_m&= \text{ max}\left(0,q^{jtn-1}_m+\alpha \left(\rho _m^{t}\left(\sum _{j=1}^Jq^{jtn-1}_m\right)+ {\partial \rho _m^{t}\left(\sum _{j=1}^Jq^{jtn-1}_m\right)\over \partial q^{jt}_m}q^{jtn-1}_m \right.\right.\nonumber \\&\qquad \qquad \left.\left.-c_{jt}-\gamma _{j}^{tn-1}\right)\right),\end{aligned}$$

  (37)
  $$\begin{aligned} {\bar{\gamma }}^{tn-1}_j&= \text{ max}\left(0,\gamma ^{tn-1}_j+\alpha \left(\sum _{m=1}^Mq_{m}^{jtn-1}-\theta _{j}\sum _{i=1}^I(q_{j1}^{itn-1}+q_{j2}^{itn-1}) \right)\right). \end{aligned}$$

  (38)

• Step 2 Adaptation Compute $X^n$ by solving the variational inequality subproblem:

  $$\begin{aligned} \left\langle ({ X}^n+(\alpha F({\bar{X}}^{n-1})-X^{n-1}))^T,X-X^n\right\rangle \ge 0, \quad \forall X\in \mathcal{K }. \end{aligned}$$

  (39)

  In particular, the explicit formulae for the solution to variational inequality (40) are as follows:

  $$\begin{aligned} {q}^{itn}_{j1}&= \text{ max}\left(0,q^{itn-1}_{j1}+\alpha \left(\theta _{j}{\bar{\gamma }_{j}}^{tn-1}-{\partial h_{ij1}^t({\bar{q}}_{j1}^{itn-1},\Pi _{t-1}^i)\over \partial q_{j1}^{it}}\right.\right.\nonumber \\&\qquad \qquad \left.\left.-\, c_{it}-{\bar{\lambda }}^{itn-1}-{\partial \hat{h}_{ij1}^t({\bar{q}}_{j1}^{itn-1}, \Pi _{t-1}^j)\over \partial q_{j1}^{it}}\right)\right),\end{aligned}$$

  (40)
  $$\begin{aligned} {q}^{itn}_{j2}&= \text{ max}\left(0,q^{itn-1}_{j2}+\alpha \left(\theta _{j}{\bar{\gamma }}_{j}^{tn-1}-{\partial h_{ij2}^t({\bar{q}}_{j2}^{itn-1},\Pi _{t-1}^i)\over \partial q_{j2}^{it}}\right.\right.\nonumber \\&\qquad \qquad \left.\left.-\,{\partial r_{ij}^t({\bar{q}}_{j2}^{itn-1},\Pi _{t-1}^j)\over \partial q_{j2}^{it}}-c_{it}-{\bar{\lambda }}^{itn-1}-{\partial \hat{h}_{ij2}^t({\bar{q}}_{j2}^{itn-1}, \Pi _{t-1}^j)\over \partial q_{j2}^{it}}\right)\right),\nonumber \\ \end{aligned}$$

  (41)
  $$\begin{aligned} {\lambda }^{itn}&= \text{ max}\left(0, \lambda ^{itn-1}+\alpha \left(\sum _{j=1}^J\left({\bar{q}}_{j1}^{itn-1}+{\bar{q}}_{j2}^{itn-1}-PCAP_{it}\right)\right)\right),\end{aligned}$$

  (42)
  $$\begin{aligned} {q}^{jtn}_m&= \text{ max}\left(0,q^{jtn-1}_m+\alpha \left(\rho _m^{t}\left(\sum _{j=1}^J{\bar{q}}^{jtn-1}_m\right)+ {\partial \rho _m^{t}\left(\sum _{j=1}^J {\bar{q}}^{jtn-1}_m\right)\over \partial q^{jt}_m} {\bar{q}}^{jtn-1}_m\right.\right.\nonumber \\&\qquad \qquad \left.\left.-\,c_{jt}-{\bar{\gamma }}_{j}^{tn-1} \right)\right),\end{aligned}$$

  (43)
  $$\begin{aligned} {\gamma }^{tn}_j&= \text{ max}\left(0,\gamma ^{tn-1}_j+\alpha \left(\sum _{m=1}^M{\bar{q}}_{m}^{jtn-1}-\theta _{j}\sum _{i=1}^I\left({\bar{q}}_{j1}^{itn-1}+{\bar{q}}_{j2}^{itn-1}\right) \right)\right). \end{aligned}$$

  (44)

1.3 Convergence verification

If $|X^n-X^{n-1}|_{\infty }\le \epsilon$, for $\epsilon >0$, a prespecified tolerance, then, stop; otherwise, set $n:=n+1$ and go to Step 1.

In the computational studies we set the scalar $\alpha =0.05$ and the tolerance $\epsilon =0.00001$.