Financial Model Generalised Input-Output Models

The type of input-output model used to model the emissions from the UK, is what Miller & Blair (2009) characterised as a “generalised product by industry input-output model”, which includes capital expenditure. In this instance, the term “product by industry” means that the model is derived from the quantities of each product sector output used by each industry sector to produce one unit of output. As each product and industry sector are comprised of heterogeneous products and industries, the amounts of products used by industry sectors are quoted in financial terms. Capital expenditures (Gross Fixed Capital Formation) are normally part of final demand but capital may be expended to provide e.g. machinery or premises that are used in the manufacture of goods or provision of services. This capital requires replenishment on timescales longer than the year which is used for intermediate demand, but it presents a more complete picture of the resources used in an economy to take the use of capital into account. Balanced against that, capital may contribute to several years of production so allocating it in one year overstates its impact on production. The sole exclusion is that capital used to replenish housing is not included in the derivation of the model. The inclusion of GFCF is common in this type of model. For example, Lenzen (2001b) argues that all production is for consumption and that so the formation of capital should be included along with intermediate consumption. However, Peters and Hertwich (2004) point out that there are two categories of capital: that which is intended to replace goods worn out in use and also includes production of goods for stock (inventory); and that which is used to finance expansion of an industry. The former could be regarded as part of production but being used outside the timescales of intermediate consumption, which is goods consumed in one year. Therefore, in estimating impacts of production it is reasonable to include this element. Expansion of capital stock to increase

production capacity may not reflect the actions of a year to year change in final demand and so should be excluded (Peters and Hertwich, 2004). In this thesis, it is assumed that both uses of capital should be included in assessing the impacts of production and hence the financial model incorporates all GFCF.

Turning to the specifics, an EEIO model based upon the UK national accounts is outlined below. In the discussion that follows a bold capital letter (e.g. A)is used to denote a matrix, a bold lowercase letter (e.g. y) is used to denote a vector. The element of a vector or matrix will be denoted by italic lowercase with subscripts representing the row and column respectively of the element within the matrix, such that 𝑎𝑖𝑗 represents the jth element of the ith row of A.

Table 8 Table of variables Used in IO Model (BP = valued at Basic Prices, PP=valued at purchaser’s prices)

Label Description Type Units

𝒙𝑫 n x 1 vector of Domestically produced output Data (ONS) £(BP)

𝒁𝑫 n x n matrix of Intermediate Industrial Consumption

Data (ONS) £(PP)

𝑲𝑫 n x n matrix of Gross Fixed Capital Formation Data (ONS) £(PP)

𝒚𝑫 n x 1 vector of domestic final demand Data (ONS) £(PP)

𝒚_𝑯𝑯𝑫 n x 1 vector of final consumption by Households Data (ONS) £(PP)

𝒚𝑵𝑷𝑰𝑺𝑯𝑫 n x 1 vector of final consumption by NPISH Data (ONS) £(PP)

𝒚𝑹𝑮𝑫 n x 1 vector of final consumption by Regional Government

Data (ONS) £(PP)

𝒚_𝑪𝑮𝑫 n x 1 vector of final consumption by Central Government

Data (ONS) £(PP)

𝒚_{𝑮𝑭𝑪𝑭}𝑫 n x 1 vector of Gross Fixed Capital Formation (GFCF)

Data (ONS) £(PP)

𝒚𝑽𝑫 n x 1 vector of changes in Valuables Data (ONS) £(PP)

𝒚𝑰𝑫 n x 1 vector of changes in Inventories Data (ONS) £(PP)

𝒚_{𝒆𝒙 𝑮𝑭𝑪𝑭}𝑫 n x 1 vector of final domestic demand less GFCF Data (ONS) £(PP)

𝑨𝑫 _{n x n matrix of technical coefficients of the} Industry intermediate consumption per unit of domestically produced output 𝒙𝑫

Parameter calculated from Data (ONS)

£(PP)/£(BP)

𝑩𝑫 n x n matrix of sectoral flows of fixed capital per unit of domestically produced output 𝒙𝐷

Parameter calculated from Data (ONS)

£(PP)/£(BP)

𝒊 𝑛×1 summation vector which on post multiplication of a 𝑛×𝑛 matrix gives a 𝑛×1 vector where the 𝑖th entry is the total of the 𝑖th row of the matrix

Summation vector

Consider the (𝑛×1) vector 𝒙𝑫(see Table 8 for definitions of variables in the following) of the domestic output of the UK economy in n sectors, this can be related to the (𝑛×𝑛) matrix 𝒁𝑫 of Intermediate Industrial Consumption and the (𝑛𝑥1) vector 𝒚𝑫 of domestic final demand from the 𝑛

sectors of the UK economy then the output of the economy can be related to the other terms using a summation vector 𝒊:

𝒙𝑫 = 𝒁𝑫𝒊 + 𝒚𝑫 (3.1)

that is the domestic output of the 𝑖th sector of the UK economy is given by

∑

𝑛_𝑗=1

𝑧

_𝑖𝑗𝐷

+ 𝑦

_𝑖𝐷 the sum of the intermediate industrial consumption and final consumption.

We then assume that intermediate industrial consumption is a linear function of 𝒙𝑫using the second principle of national accounting of proportionality, and introducing a (𝑛×𝑛) matrix of technical coefficients 𝑨𝑫, which is the direct requirements of each industry in purchasers prices per pound sterling of output in basic prices.

𝒁𝑫𝒊 = 𝒇(𝒙𝑫) = 𝑨𝑫𝒙𝑫 (3.2)

so substituting in equation (3.1) gives,

𝒙𝑫 = 𝑨𝑫𝒙𝑫+ 𝒚𝑫 (3.3)

,

𝒚𝑫= 𝒚𝑯𝑯𝑫 + 𝒚𝑵𝑷𝑰𝑺𝑯𝑫 + 𝒚𝑳𝑮𝑫 + 𝒚𝑪𝑮𝒅 + 𝒚𝑮𝑭𝑪𝑭𝑫 + 𝒚𝑽𝑫+ 𝒚𝑰𝑫 (3.4)

We assume that the Gross Fixed Capital Formation is also a linear function of 𝒙𝑫, introducing a

(𝑛×𝑛) matrix of GFCF coefficients 𝑩𝑫, which is the direct GFCF requirement in purchaser’s prices per unit of financial output in basic prices.

𝒚𝑮𝑭𝑪𝑭𝑫 = 𝒇(𝒙𝑫) = 𝑩𝑫𝒙𝑫 (3.5)

and then using 𝒚𝒆𝒙 𝑮𝑭𝑪𝑭𝑫 to gather the other elements of final demand, 𝒚𝑫can be written as

𝒚𝑫_{= 𝑩}𝑫_𝒙𝑫_{+ 𝒚} 𝒆𝒙 𝑮𝑭𝑪𝑭

𝑫 _(3.6)

Substituting for 𝒚𝑫 _{from (3.5) into (3.4),}

𝒙𝑫 = 𝑨𝑫𝒙𝑫+ 𝑩𝑫𝒙𝑫+ 𝒚𝒆𝒙𝑮𝑭𝑪𝑭𝑫 (3.7)

Collecting terms in 𝒙𝑫 _{on the LHS and factorising,}

(𝑰 − (𝑨𝑫+ 𝑩𝑫)) 𝒙𝑫 = 𝒚𝒆𝒙 𝑮𝑭𝑪𝑭𝑫 (3.8) Pre-multiplying by the inverse of (𝑰 − (𝑨𝑫+ 𝑩𝑫)) 𝒙𝑫,

𝒙𝑫 = (𝑰 − (𝑨𝑫 + 𝑩𝑫))−1𝒚𝒆𝒙 𝑮𝑭𝑪𝑭𝑫 (3.9)

Where (𝑰 − (𝑨𝑫 _{+ 𝑩}𝑫₎₎−1 _{is referred to as the Leontief inverse 𝑳 (Miller and Blair, 2009)and} summarises the total requirements of intermediate demand and GFCF over all industries in coefficients that are the total amount in purchaser’s prices of each product per unit of output in basic prices of each industry. Rewriting our equation using the Leontief inverse, 𝑳 = (𝑰 − (𝑨𝑫+ 𝑩𝑫))−𝟏,

𝒙𝑫 = 𝑳𝒚_{𝒆𝒙 𝑮𝑭𝑪𝑭}𝑫 (3.10)

The outcome of this algebraic manipulation is the relation of output of the economy 𝒙𝑫 to domestic final demand 𝒚_{𝒆𝒙 𝑮𝑭𝑪𝑭}𝑫 _{by a constant matrix 𝑳. By the assumption of proportionality, we can} calculate the output 𝒙∗ or change in output 𝚫𝒙∗, in financial terms, that arises from a final demand

A subtlety of the model is that by our inclusion of fixed capital in the Leontief inverse, the final demand element represented by GFCF should not be included when using this model. We now have the tools to estimate the financial impact of changes in final demand, but for Carbon foot printing we need a means of connecting financial impacts with GHG emissions. In the next section, we consider the emissions that are associated with industry sectors and calculate the quantity of emissions per unit output.

In document Quantifying greenhouse gases in business supply chains (Page 61-66)