Fashion Supply Chain Network Competition with Ecolabeling

Abstract

In this chapter we develop a competitive supply chain network model for fashion that incorporates ecolabeling. We capture the individual profit-maximizing behavior of the fashion firms which incur ecolabeling costs with information associated with the carbon footprints of their supply chains revealed to the consumers. Consumers, in turn, reflect their preferences for the branded products of the fashion firms through their demand price functions, which include the carbon emission information. We construct the underlying network structure of the fashion supply chains and provide alternative variational inequality formulations of the governing Nash equilibrium conditions. The model, as a special case, also captures carbon taxes. We discuss qualitative properties of the equilibrium product flow pattern and also propose an algorithm, which has elegant features for computational purposes. We provide both an illustrative example as well as a variant and then discuss a case study with several larger numerical examples.

Appendix

Proof

Proof of Theorem 1

Variational inequality (4.12) follows directly from Gabay and Moulin (1980); see also Dafermos and Nagurney (1987). We now observe that,

$$\nabla_{X_i} \hat U_i(X) = \left[\frac{\partial \hat U_i}{\partial x_p}; p\in P^i_k; k=1,\dots,n_R\right],$$

(A.1)

where for each path p; $p\in P^i_k$,

$$\begin{aligned} \frac{\partial \hat U_i}{\partial x_p} =& \frac{\partial \left[\sum_{j=1}^{n_R} \rho_{ij}(d, E)d_{ij} - \sum_{b\in L^i} \hat c_b(f, e_b(f_b))-l_i(\sum_{j=1}^{n_R} d_{ij})\right]}{\partial x_p} \nonumber\\ =& \sum_{j=1}^{n_R} \frac{\partial \left[\rho_{ij}(d, E)d_{ij}\right]}{\partial x_p}-\sum_{b\in L^i} \frac{\partial \hat c_b(f, e_b(f_b))}{\partial x_p}-\frac{\partial l_i(\sum_{j=1}^{n_R} d_{ij})}{\partial x_p} \nonumber\\ =& \sum_{j=1}^{n_R}\sum_{l=1}^{n_R} \frac{\partial\left[\rho_{ij}(d, E)d_{ij}\right]}{\partial d_{il}}\frac{\partial d_{il}}{\partial x_p}+\sum_{j=1}^{n_R} \frac{\partial \left[\rho_{ij}(d, E)d_{ij}\right]}{\partial E_i}\frac{\partial E_i}{\partial x_p} \nonumber\\ &-\sum_{a\in L^i}\sum_{b\in L^i} \frac{\partial \hat c_b(f, e_b(f_b))}{\partial f_a}\frac{\partial f_a}{\partial x_p}-\sum_{l=1}^{n_R}\frac{\partial l_i(\sum_{j=1}^{n_R} d_{ij})}{\partial d_{il}}\frac{d_{il}}{\partial x_p} \nonumber\\ =& \sum_{j=1}^{n_R} \frac{\partial\left[\rho_{ij}(d, E)d_{ij}\right]}{\partial d_{ik}}+\sum_{j=1}^{n_R} \frac{\partial \left[\rho_{ij}(d, E)d_{ij}\right]}{\partial E_i}\frac{\partial \left[\sum_{a\in L^i} e_a(f_a)\right]}{\partial x_p} \nonumber\\ &-\sum_{a\in L^i}\sum_{b\in L^i} \frac{\partial \hat c_b(f, e_b(f_b))}{\partial f_a}\delta_{ap}-\frac{\partial l_i(\sum_{j=1}^{n_R} d_{ij})}{\partial d_{ik}} \nonumber\\ =& \rho_{ik}(d,E)+\sum_{j=1}^{n_R} \frac{\partial \rho_{ij}(d, E)}{\partial d_{ik}}d_{ij}+\sum_{j=1}^{n_R} \frac{\partial \rho_{ij}(d, E)}{\partial E_i}d_{ij}\sum_{a\in L^i} \frac{\partial e_a(f_a)}{\partial f_a}\frac{\partial f_a}{\partial x_p} \nonumber\\ &-\sum_{a\in L^i}\sum_{b\in L^i} \frac{\partial \hat c_b(f, e_b(f_b))}{\partial f_a}\delta_{ap}-\frac{\partial l_i(\sum_{j=1}^{n_R} d_{ij})}{\partial d_{ik}} \nonumber\\ =& \rho_{ik}(d,E)+\sum_{j=1}^{n_R} \left[\frac{\partial \rho_{ij}(d, E)}{\partial d_{ik}}+\frac{\partial \rho_{ij}(d, E)}{\partial E_i}\sum_{a\in L^i} \frac{\partial e_a(f_a)}{\partial f_a}\delta_{ap}\right]d_{ij} \nonumber\\ &-\sum_{a\in L^i}\sum_{b\in L^i} \frac{\partial \hat c_b(f, e_b(f_b))}{\partial f_a}\delta_{ap}-\frac{\partial l_i(\sum_{j=1}^{n_R} d_{ij})}{\partial d_{ik}}.\end{aligned}$$

(A.2)

By using the conservation of flow equation (4.1) and the definitions in (4.14), (4.15), and (4.16), variational inequality (4.13) is immediate. In addition, the equivalence between variational inequalities (4.13) and (4.17) can be proved with (4.1) and (4.3).▪

We now provide some qualitative properties of the equilibrium solution. Since the feasible set K ¹ is not compact, we cannot obtain the existence of a solution simply based on the assumption of the continuity of F. However, the demand d _ik for each fashion firm i’s product, $i=1,\dots,I$ at every demand market R _k ; $k=1,\dots,n_R$, may be assumed to be bounded by the market size. Consequently, in light of (4.1), we have,

$${\cal K}_b\equiv \{x\vert \, 0\le x\le b,\},$$

(A.3)

where $b>0$ and $x\le b$ means that $x_p\le b$ for all $p\in P^i_k$; $i=1,\dots,I$ and $k=1,\dots,n_R$. Then ${\cal K}_b$ is a bounded, closed, and convex subset of K ¹. Thus, the following variational inequality

$$\langle F(X^b), X-X^b\rangle \ge 0, \forall X\in{\cal K}_b,$$

(A.4)

admits at least one solution that $X^b\in{\cal K}_b$, since ${\cal K}_b$ is compact and F is continuous. Therefore, following Kinderlehrer and Stampacchia (1980;see also Nagurney 1999), we have Theorem 2.

Theorem 2

Existence

There exists at least one solution to variational inequality ( 4.13 ) (equivalently, ( 4.17 ) ), since there exists a $b> 0$ , such that variational inequality ( A.4 ) admits a solution in ${\cal K}_b$ with

$$x^b\le b.$$

(A.5)

Furthermore, we study the uniqueness of the equilibrium solution in Theorem 3.

Theorem 3

Uniqueness

With Theorem 2, variational inequality ( A.4 ) and, hence, variational inequality ( 4.17 ) admits at least one solution. Moreover, if the function F(X) of variational inequality ( 4.17 ) , as defined in ( 4.20 ) and ( 4.21 ), is strictly monotone on ${\cal K}\equiv K^2$ , that is,

$$\langle F(X^1)-F(X^2), X^1-X^2\rangle> 0, \forall X^1,X^2\in{\cal K},\, X^1\ne X^2,$$

(A.6)

then the solution to variational inequality ( 4.17 ) is unique, that is, the equilibrium link flow pattern and the equilibrium demand pattern are unique.