Connected substitutes and invertibility of demand (Q2871449)

The authors consider a general setting in which the demand for goods \(1,\dots,J\) is characterized by \(\sigma(x)=(\sigma_1(x),\dots,\sigma_{J}(x)):\;{\mathcal X}\subset\mathbb R^{J}\to\mathbb R^{J}\), \(\sigma_0(x)=1-\sum_{j=1}^{J}\sigma_{j}(x)\), where \(x=(x_1,\dots,x_{J})\) is a vector of demand shifters. All other arguments of the demand system are held fixed. This paper deals with the invertibility of the considered demand system. Let us consider the directed graph \(\Sigma(x)\) which has nodes representing each good and a directed edge from node \(k\) to node \(l\) whenever good \(k\) substitutes to good \(l\) at \(x\).NEWLINENEWLINEThe authors prove that if: 1) \({\mathcal X}\) is a Cartesian product; 2) (weak substitutes) \(\sigma_{j}(x)\) is weakly decreasing in \(x_{k}\) for all \(j\in \{0,1,\dots,J\}\), \(k\notin \{0,j\}\); 3) (connected strict substitution) for all \(x\in {\mathcal X}^{*}\subseteq {\mathcal X}\), the directed graph \(\Sigma(x)\) has, from every node \(k\neq 0\), a directed path to node \(0\), then \(\sigma\) is inverse isotone on \({\mathcal X}^{*}\subseteq {\mathcal X}\).NEWLINENEWLINEIf \(\sigma(x)\) is differentiable on \({\mathcal X}^{*}\subseteq {\mathcal X}\) and assumption 2) (weak substitutes) holds, then the following conditions are equivalent: a) the Jacobian matrix \({\mathbf J}_{\sigma}(x)=\{\partial \sigma_{i}/\partial x_{j}\}\), \(i,j=1, \dots, J\), is nonsingular on \({\mathcal X}^{*}\); b) \({\mathbf J}_{\sigma}(x)\) is a \(P\)-matrix on \({\mathcal X}^{*}\); c) for all \(x\in {\mathcal X}^{*}\) and any nonempty \({\mathcal K}\subset\{1,\dots,J\}\), there exist \(k\in{\mathcal K}\) and \(l\notin{\mathcal K}\) such that \( \partial \sigma_{l}/\partial x_{k}<0\).