Young Duality for Variational Inequalities and Nonparametric Method of Demand Analysis in Input–Output Models with Inputs Substitution: Application for Kazakhstan Economy

The global macroeconomic shocks of the last decade entail the restructuring of national production networks and induce processes of input substitution. We suggest mathematical tools of Young duality for variational inequalities for studying these processes. Based on the tools we provide, a new mathematical model of a production network with several final consumers is created. The model is formulated as a pair of conjugated problems: a complementarity problem for optimal resource allocation with neoclassical production functions and the Young dual problem for equilibrium price indices on network products. The solution of these problems gives an equilibrium point in the space of network inter-industry flows and price indices on goods. Based on our previous results, we suggest an algorithm for model identification with an official economic statistic in the case of constant elasticity of substitution production functions. We give an explicit solution to the complementarity problems in this case and develop the algorithm of the inter-industry flows scenario projection. Since the algorithm needs the scenario projection of final sales structure as its input, we suggest a modified methodology that allows the calculation of scenario shifts in final consumer spending. To do this, we employ the generalized nonparametric method of demand analysis. As a result, we develop new technology for scenario calculation of a national input–output table, including shifts in final consumer spending. The technology takes into account a substitution of inputs in the network and is based on officially published national statistics data. The application of the methodology to study tax collection scenarios for Kazakhstan’s production network is demonstrated.