Input-output analysis with a sphere packing approach
Input-output analysis with linear programming is widely used in the planning of national economies. However, linear programming attains its optimum at a vertex of the feasible set, and a small deviation of the optimal plan in some direction violates a constraint. So far, little attention has been paid to finding sets of production plans which guarantee both the supply-demand balance of an economy and a given level of its gross domestic product (GDP), and the sphere packing approach cannot be applied directly, since the balance equations define a set with an empty interior. In this paper, we introduce the set of $(\varepsilon,δ)$-solutions, in which GDP may be smaller than its optimal value by at most $\varepsilon$ and supply may differ from demand by at most $δ$, and we find a sphere with the maximum radius inscribed into this set by solving a linear programming problem. We prove that the optimal radius is positive for positive tolerances, that it is non-decreasing, concave and piecewise linear with respect to the tolerances, and that it is bounded by the narrowest balance constraint, which is determined by a sales-concentration index of sectors. The approach is applied to the official input-output tables of Mongolia for 2010-2019 and 2024. The classical optimal plan has zero robustness, while the robust plan keeps 99.73% of the maximal GDP and, in a Monte Carlo simulation, tolerates independent deviations of all outputs and final demands of up to $\pm$0.30%. The margin of robust plans is limited by sectors with concentrated sales, mainly education and, increasingly, export-oriented mining, which indicates that a diversification of the users of mining output widens the set of robust plans.