Graphical models simplify the analysis of multivariate observations by summarizing conditional independences in the data. Variables are represented by nodes, and the absence of an edge between two nodes signifies their conditional independence. While graphical modelling has been used in several...

In a pure exchange economy, there exists a price vector which is a quasi-equilibrium (Debreu 1962), but this may not be a competitive equilibrium if some individuals' demand functions are discontinuous because their incomes may be zero. We show nonetheless, in a pure exchange economy with free...