comparable in size to (and in many cases greater than) the production emissions in many individual countries.

Calculating the balance of trade in virtual carbon has been focal to many of these contributions. In this context it is worth noting that the broader idea of an ‘ecological balance of payments’ is not new. Atkinson and Hamilton (2003), for example, use international trade data in an IO-type framework to derive the demand for natural resource rents attributable to any given country’s final consumption (see also Proops and Atkinson, 1998, Proops _et al_ . 1999 and Bailey and Clark, 2000, in a computable general equilibrium, or CGE, setting).<sup>3,4</sup> But arguably more important are the broader policy implications of this work. For Peters and Hertwich (2008), this is envisaged in terms of the carbon leakage problem and the construction of consumption-based carbon inventories as a building block in the response to this challenge (see, for example, Peters and Hertwich, 2008). Similarly, Helm _et al_ . (2007) use their analysis to recast the story of the UK’s carbon emissions record since 1990 (see, in addition, Wiedmann _et al_ . 2008).

To date few studies appear to have linked the results of the type of IO analysis described above to specific types of carbon reduction policy such as carbon taxation (although see Delgado, 2008, which looks at the likely burden which might fall on the EU’s traded goods sectors in this context). Nevertheless, a number of papers have examined such aspects in both theory and practice (e.g. Ismer and Neuhoff, 2007; Fischer and Fox, 2009) or used CGE models to simulate the impacts of policy changes (e.g. Babiker and Rutherford, 2005; Fischer and Fox, 2009; Mattoo _et al._ 2009a, 2009b). In what follows, therefore, we discuss our findings in relation to the specific question of taxing virtual carbon at the border. Before doing this we outline the model and methods that we use to generate our results.

# **3. The Input-Output Model**

This section sketches the matrix algebra underlying the empirical results in this paper. This draws on a body of existing knowledge including Peters (2008) (but see Miller and Blair, 1985, for an earlier exposition of IO analysis in a multi-regional context). In what follows, the notation refers to the case of _m_ regions (or countries) and _k_ sectors. Superscripts indicate regions and subscripts indicate sectors. When either superscripts or subscripts have a pair of numbers, the pair should be interpreted as indicating ‘from-to’ 12 flows. So, for example, _p_ 34 indicates exports of intermediate goods from sector 3 in region 1 to sector 4 in region 2.

> 3 In other contexts, Pedersen (1993) has used an IO framework to analyse net exports of transboundary (“acid rain”) pollution in Denmark vis-à-vis the rest of the world. Young (1996) examines the relative pollution intensity of traded (export-oriented) sectors and non-traded sectors in the Brazilian economy.

> 4 Another prominent way of thinking about the links between countries in this way is the ‘ecological footprint’. This describes the extent to which a particular country (or region) is reliant on resources from elsewhere to support domestic economic activity (Rees and Wackernagel, 1994). In the carbon example, these needs can be expressed in a number of ways such as the land required to grow the equivalent biofuel. If this required area is larger than the area actually available to that country, then in this sense the country has an ecological deficit. Turner _et al_ . (2007) provide an indication of how the ecological footprints standpoint relates to the IO approach.

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