Input-output price indexes: forgoing the Leontief and Ghosh models

ABSTRACT

In input-output analysis, the Leontief and Ghosh models can be used to determine the price indexes of goods, which is convenient for analyzing inter-industry inflation. Their respective merits are debated, but both provide the same solution. We demonstrate that, contrary to common belief, it is superfluous to use the Leontief or Ghosh model to calculate price indexes: the price index vector alone satisfies the accounting identities without assuming constant coefficients. So, in contrast to the Leontief and Ghosh models, price indexes can be derived ‘instantly’, without a round-by-round process. Conducting research on price indexes deduce from the Leontief or Ghosh model becomes pointless: it suffices to study price indexes deduced from the data. We illustrate these findings with an application for France 2018. The same is demonstrated for prices with the data given in physical quantities.

KEYWORDS:

• Input-output
• price index
• Leontief
• Ghosh
• price

JEL CODES:

• D57
• C67
• D46
• E11
• B51

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 See for example: Dietzenbacher (2005), Dietzenbacher and Miller (2015), EU Science Hub (2020), Galbusera and Giannopoulos (2018), Guilhoto (2021), Grainger and Kolstad (2010), Hendrickson and Horvath (1998), Haddad et al. (2021), Johansen (1960), Kagawa (2012), Leontief (1970, 1972, 1973), Lenzen (2011), Martin and Point (2012), Miller and Blair (2022), Napoles (2012), Nebbia (2000), Nakamura and Kondo (2009), Okuyama and Santos (2014), Perese (2010), Picher and Farmer (2021), Stahmer et al. (1990), Suh (2009), Santos et al. (2014), and United Nations (2008).

2 This model is part of the long tradition of Classical Economics, where output prices depend on input prices and not on the law of supply and demand. It is not considered as operational but corresponds to the early version of input-output analysis. See: Abraham-Frois and Lendjel (2006), Bidard and Erreygers (2007), and Bidard et al. (2009).

3 See the section ‘Prices in an open static input-output system’ in Leontief (1986, p. 28–31). For the assumptions usually considered necessary to the formation of index prices, see Folloni and Miglierina (1994) and Seton (1993). See also Kimura (1957), Moses (1974), Pasinetti (1977), Roland-Holst and Sancho (1995), and Weale (1984). This model is important in studying, for example, the intersectoral transmission of inflation. For recent works on this issue, see Cochard et al. (2016), Llop (2020) and Przybyliński and Gorzałczyński (2022).

4 See for example: Gruver (1989); Oosterhaven (1996); Dietzenbacher (1997); de Mesnard (2009); Oosterhaven (2022).

5 In this paper, we consider only the square input-output model (Leontief, 1986; Miller & Blair, 2022; Stone, 1984; Weale, 1984). Rectangular models (Mahajan et al., 2018; Miller & Blair, 2022; Stone, 1961; Suh et al., 2010) are another story.

6 The interest of a physical input-output table such as $\overline{\mathbf{Z}}$ is not its practicality and the purpose of this physical model is largely theoretical: it is not able to easily generate applications due to the heterogeneity of commodities and because there are potentially billions of goods, that is, billions of rows and columns.

7 Even if matrix $\overline{\mathbf{Z}}$ is a physical matrix, it has nothing to do with what environmental studies call a ‘physical input-output table’ where all the data are expressed in the same physical unit, generally in tons, and where the intermediate matrix is homogeneous per row and per column (see: Abraham-Frois and Lendjel (2006), Bidard and Erreygers (2007), Bidard et al. (2009), Duchin (2009), Giljum and Hubacek (2009), Konijn et al. (1997), Miller and Blair (2022), Pedersen (1999), Spulber (1964), Spulber and Dadkhah (1975), Stahmer (2000), Suh (2004), and Weisz and Duchin (2006)).

8 The vector $\overline{\mathbf{v}}$ is also called the value added vector because, as there is no profit in the Leontief model, the concept mixes labor and profit. However, as we have a price $\mathbf{w}$, it is difficult to discuss the price of profit. This is why we consider the idea of labor, but it does not change the reasoning.

9 Leontief developed his model in the continuation of the Classics. This is why Leontief (1986, pp. 19–21) considered first data in physical units – as Potron (see: Abraham-Frois and Lendjel (2006), Bidard and Erreygers (2007), and Bidard et al. (2009)). However, he understood that building a matrix in physical units could be very difficult:

Although in principle the inter-sectoral flows as represented in an input-output table can be thought of as being measured in physical units, in practice most input-output tables are constructed in value terms (Leontief, 1986, p. 21)

Thus, he switched from the physical table to the value table by assigning prices to goods:

Table 2-2 [the table in value] represents a translation of Table 2-1 [the physical table] into value terms on the assumption that the price of agricultural products is $2 per bushel, the price of manufactured goods is $5 per yard, and the price of services supplied by the household sector is $1 per man-year. (Leontief, 1986, p. 21)

Nevertheless, Leontief says later in his article:

All figures in Table 2-2 [the value table] – except the column sums in the bottom row [the total input of each sector, per column] – can also be interpreted as representing physical quantities of the goods or services to which they refer. This only requires that the physical unit in which the entries in each row are measured be redefined as being equal to that amount of output of that particular sector that can be purchased for $1 at prices that prevailed during the interval of time for which the table was constructed. (Leontief, 1986, p. 22)

Here is a ‘unitary price assumption.’ However, this assumption is flawed: Leontief clearly saw the flaw when he said that ‘except the column sums in the bottom row’ in the above excerpt: the total per column is meaningless for a physical table. Moreover, the inputs cannot have prices ($2, $5, $1 in Leontief's example) and, at the same time, all be equal to $1. This is why considering price indexes (presented below) is a better approach than considering that values represent physical quantities via the unitary price assumption: setting unitary price indexes (i.e. prices are fixed) is much better if we want to consider that values as values represent physical quantities.

10 Regarding traps in including price indexes in input-output models and the hidden assumptions behind the inclusion of price indexes, see de Mesnard (2016): the Leontief value model is coherent if and only if the inter-industry matrix of direct and indirect quantities of labor is stable over time.

11 Note that there is also an implicit assumption of synchronism between sectors, that can be relaxed (de Mesnard, 1992).

12 See: Gale (1989), p. 300; Weale (1984), p. 44; de Mesnard (2004), pp. 127–129.

13 If we consider a series of terms ${\mathbf{A}}_0^{(t)}$ each different, i.e. ${\mathbf{A}}_0^{(t)}\ne {\mathbf{A}}_0^{(t+1)}$ for any t, we cannot determine the value of their sum $\sum _{t=0}^T{\mathbf{A}}_0^{(t)}$ and if it is convergent; in short, we cannot say anything about the solution.

14 This paper is not the place to enter into the debated interpretation of the Ghosh model.

15 We do not enter into the technicalities between the production of sectors and marketable production.

16 We use a differentiated price index among sectors for labor to be more general, but this assumption is realistic: actually, as the labor market is not perfect, there is no uniformity between labor prices and price indexes among sectors. However, it would also be possible to consider the more particular case where the price indexes of labor are uniform without any difficulty.

17 Besides, it could be possible to conduct Monte-Carlo experiments, but it is not necessary here.

18 In physical terms, Leontief's term ‘technical coefficients’ is usually replaced by ‘production coefficients’; the term ‘input coefficients’ is also used. They may exceed one.

19 This may look like the equation of the Ghosh model in value terms but this is not the case (de Mesnard, 2009).