The paper describes a Boolean linear programming model to deal with the problem of selecting several economic sectors to be shutdown. The objective function is linear and the constraints are linear inequalities related to the Leontief’s input-output table.The model permits to analyze the feasibility of national economic system in which some elements of the input-output table are set
equal to zero.
In "On the structure of linear models" at the begining of paragraph X-the Topology of linear systems, Robert Solow wrote properties of being decomposable or indecompasable, acyclic or cyclic are topologica in nature. He means that indecomposablility is basically a property of connectedness.
As a consequence, in order to test if a square matrix is decomposable, one can replace its aij elements by 1 and 0, this technique is in use in the analysis of networks.This is what I done to investigate the impact originated by the interruption of several activity as a consequence of a government measure adopted to reduce the risk of contagion.
On the other hand, the model can be of support in assigning priorities to sectors which can be
gradually reopened as the spread of the virus decreases. As observed, a large Lagrange multiplier corresponds to a sector which is relevant in terms of its interdependencies.
The model recommends for opening before the closed sectors with large Lagrange multipliers.
The present study distinguishes for two aspects. First, linear programming techniques have been widely used in Input-Output model: from the primary and dual formulation of the problem to the analysis of Lagrange multipliers in order to assess the mutual relative importance of sectors.
Second, the impact of lockdown is measured in terms of annihilated interdependencies rather than in terms of canceled economic values such as revenues and monetary exchanges. For this reason, the proposed model takes into account the adjacency matrix got from input-output Table rather than the Table itself. This analysis is static in character (it is a static I-O model) because it does not take care explicitly of time, quantity and price reactions, consumption and production lags, growth of final demand (these ones are known as dynamic I-O models). The study simply considers the interruption of several activity as a consequence of a government measure adopted to reduce the risk of contagion.
The proposed model answers to the basic question if the annihilation of N sectors is feasible, secondary the model identifies in the Lagrange multipliers the means to sequencing the sectors to be reopened.