## Johannes Többen

*NTNU Industrial Ecology Programme, Trondheim, Norway*

After I comleted my studies in Economics at University of Osnabrück in 2011, I worked as a PhD researcher at the Institut of Energy and Climate Research at Froschungszentrum Jülich until 2016. My PhD thesis was supervised at the University of Groningen, the Netherlands. Since 2017, I work at NTNU Trondheim. During my PhD research I developed methodologies for estimating commodity flows under limited information and constructed a Multiregional Input-Output (MRIO) table for Germany's 16 federal states, which. In 2013, I stayed two month as a guest researcher at the Centre for Integrated Sustainability Analysis, School of Physics University of Sydney, Australia. I used the MRIO to assess the regional economic impacts of natural disasters, as well as the regional and social distributive effects of the German energy policy.

Large Multiregional Input-Output(MRIO) databases mapping flows measured in various units such as currency, tons or caloric values are the backbone of many recent environmental-economic studies. Their construction typically requires combining large amounts of partial information in a series of successive steps. These include the estimation of unobserved flows, transformations between units, handling aggregation re-classification and, finally, reconciling estimates with mass, financial and/or energy balances. The assembling of virtually all MRIO databases involves assumptions such as proportional allocation at some point, but their impact on the final database and, hence, on modelling outcomes is impossible to trace back due to the stepwise nature of the construction process.

In a recent contribution, Rodrigues et al. (2016) propose to transform Input-Output models into a network representation. Instead of estimating unknown arcs (i.e. flows that result from proportional allocation) connecting nodes of origin (sector in region )and destination (sector in region ) directly, virtual nodes are introduced that connect one node to another through known arcs (i.e.empirical). They show that this transformation yields exactly the same results but offers several advantages over the traditional representation as MRIO tables in a wide variety of contexts. These are, firstly, increased transparency through offering a direct link between inputs in terms of empirical data and modelling outcomes, secondly, increased modularity for linking data and/or models at different levels of resolution and/or using different units of measurement, and, thirdly, the reduction of data storage and computation requirements, since the number of unknown arcs is typically much larger than the number of data points. However, the important issue balancing inconsistent empirical data and possible advantages of their network approach in this context are only briefly discussed.

In this paper, we argue that the transformation into the network representation of MRIO data and models has many conceptual similarities to the primal-dual relationship of non-linear programming models typically used for the balancing of inconsistent data. We further generalize the maximum entropy model for the simultaneous estimation data in various units of measurement, levels of aggregation and possibly mismatching classifications developed in Többen (2017) and derive its corresponding unconstraint dual. We show that in this way any MRIO can be expressed as a set empirical data points and a set of corresponding Langrangian multipliers(one for each data point) of the optimal solution. In this setup, interrelations between data points (i.e. the nodes) are specified by means of concordance matrices providing qualitative information on the existence of unknown arcs between them, while the strength of these interrelations is indirectly determined through the size of Lagrangians relative to each other. Assumptions and other non-sample information can be introduced by means of priors. We find that this representation of MRIOs not only offers a transparent and flexible approach, but also offers significant advantages in terms of computational requirements.