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## The construction of input–output coefficients

It is not easy to transform the input and output tables "produced" by statistical offices into matrices of input-output coefficients. There are commodity-by-commodity and industry-by-industry input-output matrices and each of them can be constructed using different models. This chapter provides a unifying framework for all these alternatives and discusses the theoretical and practical pros and cons of the alternatives in a way that consolidates the vast literature. The chapter is authored by an expert who combines statistical office experience and academic contributions to the interface of input-output statistics and economic-environmental modeling.

## Dynamic input–output analysis

In dynamic input-output analysis investment meets the capital requirements of output growth. The model is linear and the proportionality between type i capital requirements and output j is represented by a capital coefficient. This chapter presents the dynamic input-output model, its solution, and two main issues, namely singularity of the matrix of capital coefficients and causal indeterminacy. Singularity is a mathematical problem that has been solved. Causal indeterminacy is the incompatibility between non-negative output solutions and arbitrary initial conditions, an issue related to the instability of the model. Alternative modifications of the model address the issue. The dynamic input-output model revives in three areas. Human capital formation is modeled to explain endogenous growth. Environmental accounts are added to analyze the depletion of nonrenewable resources. And lagged production and expenditure models are employed in disaster impact analysis.

## Early days of the input–output table

Input-output analysis was invented by Wassily Leontief, who continued to be closely involved with its development, mostly in the United States. This chapter organizes the history in a nice, concrete way, by tracing the input-output tables of the USA from 1939 until 2007, released in 2014. The history is peppered by observations of Leontief's close collaborators Anne Carter and Clopper Almon, shared with the author. The chapter concludes with a clear discussion of the myths of input-output tables, such as consistency and purity.

## Environmental economics

The basic input-output model is first extended by differentiating industry outputs by region. The consequent interregional input-output matrix accounts for pollution footprints of final consumption, possibly even including household income effects, which further boost output and pollution. Another extension is the internalization of cleansing activities, to account for the social cost of emissions. Attempts at full integration of production and environmental accounting, following the "materials balance principle," are critically examined. Other environmental analyses follow. Water satellite accounts facilitate the analysis of water trade. Waste input-output models integrate waste creation and management options so that waste can be tracked. Energy efficiency improvements reduce costs, which in turn boosts demand for energy: the rebound effect. The rebound effect is related to the input-output multipliers that include the household consumption effects. The extension to general equilibrium analysis is introduced.

## Environmental footprints

Direct input-output coefficients reflect the effects of the delivery of goods and services in terms of produced and non-produced inputs and environmental impacts (emissions). Evaluation of the further effects of the produced inputs yields the total input-output coefficients, which thus incorporate the multiplier effects of the final delivery of goods and services. The most concrete examples of these are footprints, which trace the environmental impacts of final consumption through the direct and indirect production requirements. After presenting a short history of environmental accounting in input-output analysis, this chapter discusses five types of footprints: ecological, carbon, material, water and land footprints. The methodology of footprint analysis is Leontief inversion of the matrix of input-output coefficients, where products are differentiated by their locations. The dimension of such a matrix is the number of products times the number of regions and this analysis is called multiregional input-output analysis. Multiregional input-output analysis traces the indirect requirements of final consumption in terms of national and international outputs. Application of environmental pressure coefficients yields the footprints. This combination of multiregional input-output and environmental analyses is also called life-cycle assessment and accounts for the environmental impacts embodied in trade.

## General equilibrium analysis

The supply-use system of production is extended to a social accounting matrix by and enriched with alternative model specifications in the blocks of final demand, production, and factor markets. Domestic, import and export prices are accommodated and different composite prices face industries, households, government and the Rest of the World, accounting for taxes and other margins. These price equations second the quantity equations. Alternative macroeconomic closure rules enable the determination of equilibrium values of output and prices, either by Leontief inversion or iteration. The demand function for private consumption nests different forms, including Cobb-Douglas, translog and the Almost Ideal Demand System. Production and trade are also generalized, using constant elasticity of substitution or translog functions, including for labor and capital demand and supply. Household consumption can be extended to a dynamic framework, to explain savings. The models, from static input-output to full-fledged computable general equilibrium are presented explicitly, in a unified manner.

## Handbook of Input–Output Analysis

In this authoritative Handbook, leading experts from international statistical offices and universities explain in detail the treatment and role of input-output statistics in the System of National Accounts. Furthermore, they address the derivation of input-output coefficients for the purpose of economic and environmental modeling, the building of applied general equilibrium models, the use of these models for efficiency analysis, and the extensions to stochastic and dynamic input-output analysis. As well as revealing and exploring the theoretical foundations, the Handbook also acts as a useful guide for practitioners.