Eric C. Howe

Linear aggregation of input-output models

The aggregation of input-output models is analyzed. Three axioms are shown to characterize a simple functional form for aggregation; then the properties of the aggregated model are analyzed relative to the original model. Since an input-output model is driven by a square substochastic matrix, these results can also be viewed as facts about abstract mappings involving substochastic matrices.

The structure of distances in networks

Part of this work is concerned with determining those graphs that produce extreme rays of the cone of shortest distance matrices of undirected linear n-vertex graphs without loops. For n ≤ 4, for example, all extreme rays come from a certain class of “elementary” graphs, but for n ≥ 5 there are nonelementary extreme rays, and these are studied further. Another part of this work is concerned with the representation of a distance matrix as a set of distances among points in a normed vector space. The set of graphs (distance matrices) representable by a given norm and the set of norms (and dimensions of spaces) by which a given graph may be represented are studied. It is shown that lx is unique among lp norms in that any graph is lx representable in Rn−1. The l1 representable distance matrices are found to be just those in the cone generated by elementaries for all n. Other results on distance matrices are also included.

Expected-value norms on matrices

Abstract For any vector norm, the function that assigns to a matrix A the “average” norm of Ax is a generalized matrix norm. Certain properties of such expected-value norms on matrices are noted, as well as a motivating example for error analysis in input-output models. For the l1 vector norm an explicit formula is given with respect to averaging over the Euclidean unit ball.

Aggregation of Markov processes: Axiomatization

We give two simple axioms that characterize a simple functional form for aggregation of column stochastic matrices (i.e., Markov processes). Several additional observations are made about such aggregation, including the special case in which the aggregated process is Markovian relative to the original one.

The regional structure of the united states economy

A methodology is developed to divide an economy into regions, then is applied to the United States. These regions represent a departure from the ones currently used by the Bureau of Economic Analysis and the Bureau of the Census, both of which are criticized for being derived in anad hoc fashion. The methodology uses a multiregional input-output model, which is viewed as the simplest type of general equilibrium model containing regional detail. The optimal regions developed using this methodology are ones that minimize aggregation error.