3.2 Method
Following the methodology proposed by Jorgeson et al. (
1987) and Jorgenson and Stiroh (
2000), consider an economy with
\(n\) discinct industries. Each of them can sell its products both to final demand and intermediate demand from other industries. The expression below shows that the nominal gross output production of the
\(i\) th sector (
\({P}_{i}{Q}_{i}\)) is sold both to final demand (
\({P}_{i}{Y}_{i}\)) and to intermediate demand (
\(\sum_{j=1}^{n}{P}_{i}{Q}_{ij}\)) from all
\(j\) sectors that require the good or service produced by
\(i\) as an intermediate input to its production:
$${P}_{i}{Q}_{i}={P}_{i}{Y}_{i}+\sum_{j=1}^{n}{P}_{i}{Q}_{ij}$$
(1)
where
\({P}_{i}\) represents the selling price of the industry’s
\(i\) goods, both to final and intermediate demand. Moreover,
\({Q}_{i}\),
\({Y}_{i}\) and
\({Q}_{ij}\) are, respectively, real gross output, real final demand and real intermediate demand produced by the
\(i\) th industry. Symmetrically, consider that the gross nominal production of all
\(i\) sectors can also be described from its inputs side. It means that each sector
\(i\) yields a homogeneous good or service that requires, for its production, an intermediate input set bought from other industries
\(\sum_{j=1}^{n}{P}_{j}{Q}_{ji}\), as well as a set of rental price of capital and labour inputs, respectively, defined as
\({P}_{Ki}{K}_{i}\) and
\({P}_{Li}{L}_{i}\), as shown by the equation below:
$${P}_{i}{Q}_{i}={P}_{{L}_{i}}{L}_{i}+{P}_{{K}_{i}}{K}_{i}+\sum_{j=1}^{n}{P}_{j}{Q}_{ji}$$
(1’)
The sectoral nominal value-added (
\({P}_{i}^{V}{V}_{i}\)), or net output, is, therefore, the difference between their respective gross production and intermediate demand.
16 In our model, it is precisely equal to the sum of sectoral primary inputs expenditures, as shown by the next expression:
$${P}_{i}^{V}{V}_{i}={P}_{i}{Q}_{i}-\sum_{j=1}^{n}{P}_{j}{Q}_{ji}={P}_{{L}_{i}}{L}_{i}+{P}_{{K}_{i}}{K}_{i}$$
(2)
Equalising (
1) to (
1’), and summing up for all the
\(i\) industries, we find the definition of the economy’s gross domestic product (GDP). It can be measured both from the sum of all final demands and value-added. It is worth noting that the intermediate inputs demand and supply cancel out each other avoiding double counting.
$$\sum\limits_{{i = 1}}^{n} {P_{i} } Y_{i} = \sum\limits_{{i = 1}}^{n} {P_{i}^{V} V_{i} } = GDP$$
(1’’)
Assume that each industry’s production technology is described, in a more general form, as a sectoral production function that relates time and its inputs—both primary and intermediate—with the gross industrial product. The Hicks-neutral type of this function is:
$${Q}_{i}={Q}_{i}\left({L}_{i},{K}_{i},{X}_{ji},t\right)$$
(3)
Differentiating totally (3) with respect to time, using (
1’) and considering that a hat (^) denotes growth rate, we find the next equation that describes the
\(i\) th sector multifactor productivity growth. For the sake of notation simplicity, the sectoral inputs to gross output shares are denoted
17 by
\({\upsilon }_{Li}=\frac{{{P}_{Li}L}_{i}}{{{P}_{i}Q}_{i}}\),
\({\upsilon }_{ki}=\frac{{{P}_{Ki}K}_{i}}{{{P}_{i}Q}_{i}}\) and
\({\upsilon }_{{Q}_{ji}}=\sum_{j=1}^{n}\frac{{{P}_{j}Q}_{ji}}{{{P}_{i}Q}_{i}}\).
$${\widehat{q}}_{i}={\widehat{Q}}_{i}-{\upsilon }_{Li}{\widehat{L}}_{i}-{\upsilon }_{ki}{\widehat{K}}_{i}-{\upsilon }_{Qji}{\widehat{Q}}_{ji}$$
(4)
The term
\({\widehat{q}}_{i}\) denotes sectoral multifactor productivity
18 growth. The multifactor productivity growth—MFP growth hereafter—is defined as the difference between the growth rate of the gross product and the growth rate of the inputs, weighted by the share of the input’s value in the value of the gross product [see, e.g., Cas and Rymes (
1991)]. One of the first authors to formalise the concept of MFP
19 growth was Hulten (
1978). Note that the equation above can be written in discrete time using a Törnquist
20 or translog discrete-time approximation, where the
\(\Delta\) term is the difference between the variable in the current and previous time:
$$\Delta ln{q}_{it}=\Delta ln{Q}_{it}-\frac{({\upsilon }_{{L}_{it}}+{\upsilon }_{{L}_{it-1}})}{2}\Delta ln{L}_{it}-\frac{({\upsilon }_{{K}_{it}}+{\upsilon }_{{K}_{it-1}})}{2}\Delta ln{K}_{it}\frac{({\upsilon }_{{Q}_{jit}}+{\upsilon }_{{Q}_{jit-1}})}{2}\Delta ln{Q}_{jit}$$
(4’)
We can describe the sectoral gross output growth rate as the average mean of the growth rates of both real net output and intermediate inputs, weighted by its respective shares of the gross production. In the equation below, the term
\({\upsilon }_{Vi}\) equals to
\({\upsilon }_{{L}_{i}}+ {\upsilon }_{{K}_{i}}\).
$${\widehat{Q}}_{i}={\upsilon }_{{V}_{i}}{\widehat{V}}_{i}+{\upsilon }_{{Q}_{jit}}{\widehat{Q}}_{ji}$$
(5)
Using (
4) and (
5) and after some algebraic manipulations, it is possible to find the following expression that relates the growth rate of the sectoral value-added with the growth rate of capital stock, labour force and productivity:
$${\upsilon }_{{V}_{i}}{\widehat{V}}_{i}={\upsilon }_{{K}_{i}}{\widehat{K}}_{i}+{\upsilon }_{{L}_{i}}{\widehat{L}}_{i}+{\widehat{q}}_{i}.$$
(6)
From an aggregate point of view, the economy’s GDP is described as the sum of all sectoral values added (or amount of all sector final demand). That is, being the nominal GDP of the whole economy
\(PY\), we have that
\(PY={P}_{v}V=\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}\). We use a general function that relates the aggregated value added with the relevant inputs and time
21:
$$V=f\left(L,K,t\right).$$
(7)
When differentiating totally (
7) with respect to time, and after some algebraic manipulations, we find an expression that connects the growth rate of aggregate productivity, defined as
\(\widehat{q}\), with the growth rate of the total value added of the economy and the weighted sum of the sectorial primary inputs capital and labour:
$$\widehat{q}=\widehat{V}-\sum_{i=1}^{n}\frac{{P}_{{L}_{i}}{L}_{i}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{L}}_{i}-\sum_{i=1}^{n}\frac{{P}_{{K}_{i}}{K}_{i}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{K}}_{i}.$$
(8)
Aiming to unearth an equation that relates the productivity growth rate of the whole economy with the growth rates of sectoral productivity—the Domar aggregation—we combine (6) and (8) to obtain:
$$\widehat{q}=\sum_{i=1}^{n}\frac{{P}_{i}{Q}_{i}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{q}}_{i}$$
(9)
Expression (
9) is known as the
Domar aggregation of sectoral MFP growth. Although Domar (
1961) was the first to find this relationship formally, other authors, such as Hulten (
1978) and Jorgeson et al. (
1987) later improved it. In discrete time, we can write the expression (
9) as:
$$\Delta {\text{lnq}}=\sum_{i=1}^{n}\frac{1}{2}\left(\frac{{P}_{it}{Q}_{it}}{\sum_{i=1}^{n}{P}_{it}^{V}{V}_{it}}+\frac{{P}_{it-1}{Q}_{it-1}}{\sum_{i=1}^{n}{P}_{it-1}^{V}{V}_{it-1}}\right)\Delta ln{q}_{it}.$$
(9’)
Note that the weighted sum of sectoral MFP has the striking feature that it sums to more than unity
22 in economies with intermediate goods. The higher the participation of intermediate inputs in the economy, the higher the sum of the weightings. Regarding the
‘sum to more than the unity’ of Domar aggregation and its intuition, Jorgenson (
2018, p. 881) considers that:
“A distinctive feature of Domar weights is that they sum to more than one, reflecting the fact that an increase in the growth of the industry’s productivity has two effects: the first is a direct effect on the industry’s output and the second an indirect effect via the output delivered to other industries as intermediate inputs.”
Similarly, Oulton and O’Mahony (
1994, p. 14) explains the intuition behind the role of intermediate inputs in the aggregated productivity growth and the Domar weights behaviour:
“The intuitive justification for the sum of the weights exceeding one is that an industry contributes not only directly to aggregate productivity growth but also indirectly, through helping lower costs elsewhere in the economy when other industries buy its product”.
The Domar aggregation method establishes a link between the industrial’s level of productivity growth and aggregate productivity growth. An aggregated economy’s overall productivity may exceed the average productivity gains across sectors, given that flows of intermediate inputs among sectors contribute to total productivity growth by allowing productivity gains—or losses—in successive industries to augment one another. Moreover, an industry’s contribution to the overall productivity growth depends (besides the direct productivity growth in this sector) on the efficiency changes in the production of its intermediate inputs. To clarify the mechanism in which the direct and indirect effects above mentioned behave within the model, we substitute Eqs. (
1’) and (2) into the numerator of (9) to obtain:
$$\widehat{q}=\sum_{i=1}^{n}\frac{{P}_{{V}_{i}}{V}_{i}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{q}}_{i}+\sum_{i=1}^{n}\frac{\sum_{j=1}^{n}{P}_{j}{Q}_{ji}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{q}}_{i}.$$
(10)
Disaggregating the second term of the expression above for all sectors, we get:
$$\widehat{q}=\sum_{i=1}^{n}\frac{{P}_{{V}_{i}}{V}_{i}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{q}}_{i}+\frac{\sum_{j=1}^{n}{P}_{j}{Q}_{j1}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{q}}_{1}+\frac{\sum_{j=1}^{n}{P}_{j}{Q}_{j2}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{q}}_{2}+...+\frac{\sum_{j=1}^{n}{P}_{j}{Q}_{jn}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{q}}_{n}.$$
(11)
Note that the sum of the value-added weights, in the first term of the equation above the right-hand side, is precisely one. The terms on the right, however, depict the sectoral productivity impacts from intermediate inputs deliveries. Therefore, the weights on the right are the ones that exceed the unity considering the overall aggregation. From the equation above, it must be clear that the higher the degree of interconnection, or density of the economy in terms of intermediate inputs deliveries, the higher the potential of productivity growth augmenting given the growth of sectoral productivities.
To visualise the mechanism involved, assume that
\({\theta }_{ij}=\frac{\sum_{j=1}^{n}{P}_{j}{Q}_{ji}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}\) is the share of aggregate demand for intermediate inputs in the economy, which measures the degree of sectoral density or sectorial interconnection. Substituting
\({\theta }_{ij}\) into Eq. (
10), we find the equation below:
$$\widehat{q}=\sum_{i=1}^{n}\frac{{P}_{Vi}{V}_{i}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{q}}_{i}+\sum_{i=1}^{n}{\theta }_{ij}{\widehat{q}}_{i}$$
(12)
The term
\(\sum_{i=1}^{n}{\theta }_{ij}\) measures the degree of interconnection, or density, of the economy, since it defines the relative importance of vertical interaction of the sectors or industries. The greater the term
\({\theta }_{ij}\) is in each
\(i\) th sector, the more significant is the sectoral capability to spread productivity and to augment the sum of the whole economy due to Domar weights. Let us suppose that, for some reason, the density
\({\theta }_{ij}\) of some sector
\(i\) increases due to a more significant share of intermediate demand by the given sector in the economy’s GDP. Then, by differentiating the aggregate productivity growth with respect to
\({\theta }_{ij}\) in (
12) we have that:
$$\frac{\partial \widehat{q}}{\partial {\theta }_{ij}}={\widehat{q}}_{i}>0$$
(13)
Hence, if the sectoral productivity growth in the given sector is positive, then an increase
23 in
\({\theta }_{ij}\) leads, by itself, to a higher aggregate productivity growth,
given all sectoral productivity growth. In this vein, if the share of intermediate goods in the economy increases, the sum of Domar weights increases as well. In that case, the economy is subject to a higher density
24 that generates an augmented potential of aggregate productivity growth. Finally, using Eqs. (
4) and (
5) and summing up for all sectors, it is possible to find an expression concerning the interactions between aggregate productivity growth and economic (GDP) growth:
$$\widehat{v}=\sum_{i=1}^{n}\frac{{p}_{i}{v}_{i}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{v}}_{i}=\sum_{i=1}^{n}\frac{{p}_{i}{Q}_{i}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{q}}_{i}+\frac{{p}_{li}{L}_{i}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{L}}_{i}+\frac{{p}_{{K}_{i}}{K}_{i}}{\sum_{i=1}^{n}{P}_{i}^{V}{V}_{i}}{\widehat{K}}_{i}.$$
(14)
Thus, as shown by the above equation, the aggregate value-added growth rate can be equivalent to the weighted sum of labour, capital, and Domar aggregated productivity growth contributions.