A Framework Without Assumptions

Abstract

The measurement of productivity change (or difference) is usually based on models relying on strong assumptions such as competitive behaviour and constant returns to scale. This chapter discusses the basics of productivity measurement and shows that one can dispense with most if not all of the usual, neo-classical assumptions. Various models are reviewed and their relationships discussed. Throughout the chapter the equivalence of multiplicative and additive models, as well as the equivalence of productivity measurement and growth accounting, is highlighted. By virtue of their structural features, the various measurement models are applicable to individual establishments and aggregates such as industries, sectors, or economies.

Appendices

Appendix A: Indices and Indicators

The basic measurement tools used are price and quantity indices and indicators. Indices are ratio-type measures, and indicators are difference-type measures. What, precisely, are the requirements for good tools? The following just serves to introduce and illustrate some concepts used in the main text of this book. For a complete treatment and historical information the reader is referred to Balk (2008).

We consider two time periods and an aggregate consisting of M items (goods and services) with (unit) prices going from $p_m^0$ to $p_m^1$ and quantities going from $y_m^0$ to $y_m^1$ (m = 1, …, M). The first subsection reviews indices, and the second subsection reviews indicators. It is important to note that, though stated in terms of output, the theory surveyed below is equally applicable to other situations after appropriate modification of the number of items and the interpretation of the variables. When there are more than two time periods involved one can choose between direct (bilateral) indices/indicators or chained (multilateral) indices/indicators. The following survey is restricted to bilateral functions.

2.1.1 Indices

A bilateral price index is a positive, continuously differentiable function $P(p^1,y^1,p^0,y^0): \Re ^{4M}_{++} \rightarrow \Re _{++}$ that, except in extreme situations, correctly indicates any increase or decrease of the elements of the price vectors p ¹ or p ⁰, conditional on the quantity vectors y ¹ and y ⁰. A bilateral quantity index is a positive, continuously differentiable function of the same variables $Q(p^1,y^1,p^0,y^0): \Re ^{4M}_{++} \rightarrow \Re _{++}$ that, except in extreme situations, correctly indicates any increase or decrease of the elements of the quantity vectors y ¹ or y ⁰, conditional on the price vectors p ¹ and p ⁰. The number M is called the dimension of the price or quantity index.

The basic requirements (aka axioms) on price and quantity indices comprise (1) that they exhibit the correct monotonicity properties; (2) that they are linearly homogeneous in comparison period prices (quantities, respectively); (3) that they satisfy the Identity Test; (4) that they are homogeneous of degree 0 in prices (quantities, respectively); (5) that they are invariant to changes in the units of measurement of the commodities. Interchanging the variables p ^t and y ^t (t = 0, 1) transforms any price index into a quantity index and vice versa.

The Product Test requires that price index times quantity index equals the value ratio; that is, P(p ¹, y ¹, p ⁰, y ⁰) × Q(p ¹, y ¹, p ⁰, y ⁰) = p ¹ ⋅ y ¹∕p ⁰ ⋅ y ⁰. This requirement is especially important in the context of deflation. Deflation of a nominal value means dividing it by a price index in order to obtain a real value; that is, something that behaves as a scalar quantity. In our two-period situation, by deflating the nominal comparison period value p ¹ ⋅ y ¹ by the price index P(p ¹, y ¹, p ⁰, y ⁰) we get as real comparison period value p ¹ ⋅ y ¹∕P(p ¹, y ¹, p ⁰, y ⁰). The Product Test then implies that this can be written as p ⁰ ⋅ y ⁰ × Q(p ¹, y ¹, p ⁰, y ⁰); that is, base period nominal value inflated by a quantity index. This behaves as a scalar quantity if and only if the quantity index satisfies the basic requirements. However, that the price index satisfies the basic requirements is not at all a sufficient condition for the quantity index to do the same.

As defined, price and quantity indices are functions of prices and quantities. In practice it is usually easier to observe (or estimate) price or quantity relatives and (nominal) values or value shares. It turns out, however, that any function P(p ¹, y ¹, p ⁰, y ⁰) or Q(p ¹, y ¹, p ⁰, y ⁰), that is invariant to changes in the units of measurement, can be written as a function of only 3M variables, namely the price relatives $p_{m}^1/p_{m}^0$ or the quantity relatives $y_{m}^1/y_{m}^0$, respectively, and the values $v_m^t \equiv p_{m}^ty_{m}^t$ (m = 1, …, M;t = 0, 1). Value shares are defined as $s_m^t \equiv v_m^t/\sum _{m=1}^{M}v_m^t$ (m = 1, …, M;t = 0, 1).

2.1.1.1 The Main Formulas

The indices materializing in this book will now be listed. The reader is invited to check their properties. We start by considering the Dutot price index, defined as

$$\displaystyle \begin{aligned} P^{D}(p^1,y^1,p^0,y^0) = \sum_{m=1}^{M}p_m^1/ \sum_{m=1}^{M}p_m^0. \end{aligned} $$

(2.116)

Notice that this index does not depend on quantities. Likewise, the Dutot quantity index is defined as

$$\displaystyle \begin{aligned} Q^{D}(p^1,y^1,p^0,y^0) = \sum_{m=1}^{M}y_m^1/ \sum_{m=1}^{M}y_m^0. \end{aligned} $$

(2.117)

This index does not depend on prices. Dutot indices are not invariant to changes in the units of measurement of the items. The implication of this is that these indices can only be used when the underlying items (goods and services) can reasonably be considered as homogeneous (or interchangeable). Notice that P ^D(p ¹, y ¹, p ⁰, y ⁰) × Q ^D(p ¹, y ¹, p ⁰, y ⁰) ≠ p ¹ ⋅ y ¹∕p ⁰ ⋅ y ⁰, except in special cases. It is also interesting to notice that the implicit Dutot price index, that is, the value index divided by the Dutot quantity index, is known as the unit-value index,

$$\displaystyle \begin{aligned} \frac{p^1 \cdot y^1}{p^0 \cdot y^0}\Big/Q^{D}(p^1,y^1,p^0,y^0) = \frac{\sum_{m=1}^{M}p_m^1y_m^1/\sum_{m=1}^{M}y_m^1} {\sum_{m=1}^{M}p_m^0y_m^0/\sum_{m=1}^{M}y_m^0}. \end{aligned} $$

(2.118)

This is a ratio of mean prices (aka unit values). The main disadvantage of the unit-value index is that in general the Identity Test for price indices is violated.

We continue by considering the Laspeyres price index. As function of prices and quantities this index is defined as

$$\displaystyle \begin{aligned} P^{L}(p^1,y^1,p^0,y^0) \equiv p^1 \cdot y^0/p^0 \cdot y^0, \end{aligned} $$

(2.119)

which can be written as a function (arithmetic mean) of price relatives and (base period) value shares,

$$\displaystyle \begin{aligned} P^{L}(p^1,y^1,p^0,y^0) = \sum_{m=1}^{M}(p_m^1/p_m^0)v_m^0/\sum_{m=1}^{M}v_m^0 = \sum_{m=1}^{M}s_m^0(p_m^1/p_m^0), \end{aligned} $$

(2.120)

The Laspeyres quantity index is defined as

$$\displaystyle \begin{aligned} Q^{L}(p^1,y^1,p^0,y^0) \equiv p^0 \cdot y^1/p^0 \cdot y^0. \end{aligned} $$

(2.121)

Notice that P ^L(p ¹, y ¹, p ⁰, y ⁰) × Q ^L(p ¹, y ¹, p ⁰, y ⁰) ≠ p ¹ ⋅ y ¹∕p ⁰ ⋅ y ⁰, except in special cases.

The Paasche price index is defined as

$$\displaystyle \begin{aligned} P^{P}(p^1,y^1,p^0,y^0) \equiv p^1 \cdot y^1/p^0 \cdot y^1, \end{aligned} $$

(2.122)

which can be written as a function (harmonic mean) of price relatives and (comparison period) value shares,

$$\displaystyle \begin{aligned} P^{P}(p^1,y^1,p^0,y^0) = \left(\sum_{m=1}^{M}(p_m^0/p_m^1)v_m^1/\sum_{m=1}^{M}v_m^1\right)^{-1} = \left(\sum_{m=1}^{M}s_m^1(p_m^1/p_m^0)^{-1}\right)^{-1}. \end{aligned} $$

(2.123)

The Paasche quantity index is defined as

$$\displaystyle \begin{aligned} Q^{P}(p^1,y^1,p^0,y^0) \equiv p^1 \cdot y^1/p^1 \cdot y^0. \end{aligned} $$

(2.124)

Notice that P ^P(p ¹, y ¹, p ⁰, y ⁰) × Q ^P(p ¹, y ¹, p ⁰, y ⁰) ≠ p ¹ ⋅ y ¹∕p ⁰ ⋅ y ⁰, except in special cases. However, P ^L(p ¹, y ¹, p ⁰, y ⁰)×Q ^P(p ¹, y ¹, p ⁰, y ⁰) = P ^P(p ¹, y ¹, p ⁰, y ⁰)×Q ^L(p ¹, y ¹, p ⁰, y ⁰) = p ¹⋅y ¹∕p ⁰⋅y ⁰.

The Fisher price index is defined as the geometric mean of the Laspeyres and Paasche indices,

$$\displaystyle \begin{aligned} P^{F}(p^1,y^1,p^0,y^0) \equiv \left[P^{L}(p^1,y^1,p^0,y^0) \times P^{P}(p^1,y^1,p^0,y^0)\right]^{1/2} \end{aligned} $$

(2.125)

$$\displaystyle \begin{aligned} = \left[\frac{\sum_{m=1}^{M}s_m^0(p_m^1/p_m^0)}{\sum_{m=1}^{M}s_m^1(p_m^1/p_m^0)^{-1}}\right]^{1/2}. \end{aligned}$$

The Fisher quantity index is similarly defined as

$$\displaystyle \begin{aligned} Q^{F}(p^1,y^1,p^0,y^0) \equiv \left[Q^{L}(p^1,y^1,p^0,y^0) \times Q^{P}(p^1,y^1,p^0,y^0)\right]^{1/2}. \end{aligned} $$

(2.126)

Notice that P ^F(p ¹, y ¹, p ⁰, y ⁰) × Q ^F(p ¹, y ¹, p ⁰, y ⁰) = p ¹ ⋅ y ¹∕p ⁰ ⋅ y ⁰.

The geometric alternative to the Laspeyres price index, called GeoLaspeyres price index, is defined as

$$\displaystyle \begin{aligned} P^{GL}(p^1,y^1,p^0,y^0) \equiv \prod_{m=1}^{M}(p_m^1/p_m^0)^{s_m^0}. \end{aligned} $$

(2.127)

The GeoLaspeyres quantity index is defined as

$$\displaystyle \begin{aligned} Q^{GL}(p^1,y^1,p^0,y^0) \equiv \prod_{m=1}^{M}(y_m^1/y_m^0)^{s_m^0}. \end{aligned} $$

(2.128)

Notice that P ^GL(p ¹, y ¹, p ⁰, y ⁰) × Q ^GL(p ¹, y ¹, p ⁰, y ⁰) ≠ p ¹ ⋅ y ¹∕p ⁰ ⋅ y ⁰, except in special cases.

The GeoPaasche price index is defined as

$$\displaystyle \begin{aligned} P^{GP}(p^1,y^1,p^0,y^0) \equiv \prod_{m=1}^{M}(p_m^1/p_m^0)^{s_m^1}, \end{aligned} $$

(2.129)

and the GeoPaasche quantity index is defined as

$$\displaystyle \begin{aligned} Q^{GP}(p^1,y^1,p^0,y^0) \equiv \prod_{m=1}^{M}(y_m^1/y_m^0)^{s_m^1}. \end{aligned} $$

(2.130)

Notice that P ^GP(p ¹, y ¹, p ⁰, y ⁰) × Q ^GP(p ¹, y ¹, p ⁰, y ⁰) ≠ p ¹ ⋅ y ¹∕p ⁰ ⋅ y ⁰, except in special cases.

The geometric mean of the GeoLaspeyres and GeoPaasche price indices is called the Törnqvist price index,

$$\displaystyle \begin{aligned} P^{T}(p^1,y^1,p^0,y^0) \equiv \prod_{m=1}^{M}(p_m^1/p_m^0)^{(s_m^0 + s_m^1)/2}. \end{aligned} $$

(2.131)

The Törnqvist quantity index is defined as

$$\displaystyle \begin{aligned} P^{T}(p^1,y^1,p^0,y^0) \equiv \prod_{m=1}^{M}(y_m^1/y_m^0)^{(s_m^0 + s_m^1)/2}. \end{aligned} $$

(2.132)

Notice that P ^T(p ¹, y ¹, p ⁰, y ⁰) × Q ^T(p ¹, y ¹, p ⁰, y ⁰) ≠ p ¹ ⋅ y ¹∕p ⁰ ⋅ y ⁰, except in special cases.

An alternative to the Törnqvist indices are the Sato-Vartia indices. Here the price or quantity relatives are weighed with logarithmic mean value shares, normalized to ensure that these add up to 1. Thus, the Sato-Vartia price index is

$$\displaystyle \begin{aligned} P^{SV}(p^1,y^1,p^0,y^0) \equiv \prod_{m=1}^{M}(p_m^1/p_m^0)^{\mathit{LM}(s_m^0,s_m^1)/\sum_{m=1}^{M}\mathit{LM}(s_m^0,s_m^1)}, \end{aligned} $$

(2.133)

where LM(.) denotes the logarithmic mean.³⁸ The Sato-Vartia quantity index is defined as

$$\displaystyle \begin{aligned} Q^{SV}(p^1,y^1,p^0,y^0) \equiv \prod_{m=1}^{M}(y_m^1/y_m^0)^{\mathit{LM}(s_m^0,s_m^1)/\sum_{m=1}^{M}\mathit{LM}(s_m^0,s_m^1)}. \end{aligned} $$

(2.134)

Notice that P ^SV(p ¹, y ¹, p ⁰, y ⁰) × Q ^SV(p ¹, y ¹, p ⁰, y ⁰) = p ¹ ⋅ y ¹∕p ⁰ ⋅ y ⁰.

A second alternative are the Montgomery-Vartia indices. The price and quantity indices are defined as, respectively,

$$\displaystyle \begin{aligned} P^{MV}(p^1,y^1,p^0,y^0) \equiv \prod_{m=1}^{M}(p_m^1/p_m^0)^{\mathit{LM}(v_m^0,v_m^1)/\mathit{LM}(V^0,V^1)}, \end{aligned} $$

(2.135)

$$\displaystyle \begin{aligned} Q^{MV}(p^1,y^1,p^0,y^0) \equiv \prod_{m=1}^{M}(y_m^1/y_m^0)^{\mathit{LM}(v_m^0,v_m^1)/\mathit{LM}(V^0,V^1)}, \end{aligned} $$

(2.136)

where $V^t \equiv \sum _{m=1}^{M}v_m^t$ (t = 0, 1). Now the price and quantity relatives are weighed with ratios of means instead of means of ratios. These weights do not add up to 1, though generally the discrepancy appears to be negligible. Notice that P ^MV(p ¹, y ¹, p ⁰, y ⁰) × Q ^MV(p ¹, y ¹, p ⁰, y ⁰) = p ¹ ⋅ y ¹∕p ⁰ ⋅ y ⁰.

2.1.1.2 Two-Stage Indices

It is important to consider the concept of two-stage indices. Let the aggregate under consideration be denoted by A, and let A be partitioned arbitrarily into K disjunct subaggregates A _k,

$$\displaystyle \begin{aligned} A = \cup^{K}_{k = 1}A_{k},\: A_{k} \cap A_{k'} = \emptyset \: (k \neq k'). \end{aligned}$$

Each subaggregate consists of a number of items. Let M _k ≥ 1 denote the number of items contained in A _k (k = 1, …, K). Obviously $M = \sum ^{K}_{k=1}M_{k}$. Let $(p^{1}_{k},y^{1}_{k},p^{0}_{k},y^{0}_{k})$ be the subvector of (p ¹, y ¹, p ⁰, y ⁰) corresponding to the subaggregate A _k. Recall that $v_m ^{t} \equiv p_m^ty_m^t$ is the value of item m at period t. Then $V^{t}_{k} \equiv \sum _{m \in A_k}v^{t}_{m}$ (k = 1, …, K) is the value of subaggregate A _k at period t, and $V^{t} \equiv \sum _{m \in A}v^{t}_{m} = \sum ^{K}_{k=1}V^{t}_{k}$ is the value of aggregate A at period t.

Let P(.), P ⁽¹⁾(.), …, P ^(K)(.) be price indices of dimension K, M ₁, …, M _K respectively that satisfy the basic requirements. Recall that any such price index can be written as a function of price relatives and base and comparison period item values. Replace in P(.) the price relatives by subaggregate price indices and the item values by subaggregate values. Then the price index defined by

$$\displaystyle \begin{aligned} P^{*}(p^1,y^1,p^0,y^0) \equiv P(P^{(k)}(p^{1}_{k},y^{1}_{k},p^{0}_{k},y^{0}_{k}),V_k^1,V_k^0; k=1,\ldots ,K) \end{aligned} $$

(2.137)

is of dimension M and also satisfies the basic requirements. The index P ^∗(.) is called a two-stage index. The first stage refers to the indices P ^(k)(.) for the subaggregates A _k (k = 1, …, K). The second stage refers to the index P(.) that is applied to the subindices P ^(k)(.) (k = 1, …, K). A two-stage index such as defined by expression (2.137) closely corresponds to the calculation practice at statistical agencies. All the subindices are then usually of the same functional form, for instance Laspeyres or Paasche indices. The aggregate, second-stage index may or may not be of the same functional form. This could be, for instance, a Fisher index.

If the functional forms of the subindices P ^(k)(.) (k = 1, …, K) and the aggregate index P(.) are the same, then P ^∗(.) is called a two-stage P(.)-index. Continuing the first example, the two-stage Laspeyres price index reads

$$\displaystyle \begin{aligned} P^{*L}(p^1,y^1,p^0,y^0) \equiv \sum_{k=1}^{K}P^L(p^{1}_{k},y^{1}_{k},p^{0}_{k},y^{0}_{k})V_k^0/\sum_{k=1}^{K}V_k^0, \end{aligned} $$

(2.138)

and one simply checks that the two-stage and the single-stage Laspeyres price indices coincide; that is,

$$\displaystyle \begin{aligned} P^{*L}(p^1,y^1,p^0,y^0) = P^{L}(p^1,y^1,p^0,y^0). \end{aligned} $$

(2.139)

However, this is the exception rather than the rule. For most indices, two-stage and single-stage variants do not coincide. Put otherwise, most indices are not Consistent-in-Aggregation (CIA).

Two-stage quantity indices can be defined analogously.

2.1.2 Indicators

The continuous functions $\mathcal {P}(p^1,y^1,p^0,y^0): \Re ^{4M}_{++} \rightarrow \Re$ and $\mathcal {Q}(p^1,y^1,p^0,y^0): \Re ^{4M}_{++} \rightarrow \Re$ will be called price indicator and quantity indicator, respectively, if they satisfy the following basic requirements (aka axioms): (1) the functions exhibit the correct monotonicity properties; (2) they satisfy the Identity Test; (3) they are homogeneous of degree 1 in prices or quantities, respectively; (4) they are invariant to changes in the units of measurement of the commodities. Interchanging the variables p ^t and y ^t (t = 0, 1) transforms any price indicator into a quantity indicator and vice versa. The analogue of the Product Test requires that price indicator plus quantity indicator equals the value difference; that is, $\mathcal {P}(p^1,y^1,p^0,y^0) + \mathcal {Q}(p^1,y^1,p^0,y^0) = p^1 \cdot y^1 - p^0 \cdot y^0$. Notice that these functions may take on negative or zero values.

As values are additive, we have

$$\displaystyle \begin{aligned} V^{1} - V^{0} = \sum_{m \in A}(v^{1}_{m} - v^{0}_{m}). \end{aligned} $$

(2.140)

Given a pair of price and quantity indicators satisfying the Product Test, it must then be the case that

$$\displaystyle \begin{aligned} \mathcal{P}(p^1,y^1,p^0,y^0) + \mathcal{Q}(p^1,y^1,p^0,y^0) = \end{aligned}$$

$$\displaystyle \begin{aligned} \sum_{m \in A}\left(\mathcal{P}(p_m^1,y_m^1,p_m^0,y_m^0) + \mathcal{Q}(p_m^1,y_m^1,p_m^0,y_m^0)\right), \end{aligned} $$

(2.141)

where the dimension of the indicators at the right-hand side of this equation equals 1. A natural requirement is then that $\mathcal {P}(p^1,y^1,p^0,y^0) = \sum _{m \in A}\mathcal {P}(p_m^1,$ $y_m^1,p_m^0,y_m^0)$ and $\mathcal {Q}(p^1,y^1,p^0,y^0) = \sum _{m \in A}\mathcal {Q}(p_m^1,y_m^1,p_m^0,y_m^0)$. Any such indicator is CIA, as

$$\displaystyle \begin{aligned} \mathcal{P}(p^1,y^1,p^0,y^0) = \sum_{k=1}^{K}\mathcal{P}(p^{1}_{k},y^{1}_{k},p^{0}_{k},y^{0}_{k}) = \sum_{k=1}^{K}\sum_{m \in A_k}\mathcal{P}(p^{1}_{m},y^{1}_{m},p^{0}_{m},y^{0}_{m}), \end{aligned} $$

(2.142)

and

$$\displaystyle \begin{aligned} \mathcal{Q}(p^1,y^1,p^0,y^0) = \sum_{k=1}^{K}\mathcal{Q}(p^{1}_{k},y^{1}_{k},p^{0}_{k},y^{0}_{k}) = \sum_{k=1}^{K}\sum_{m \in A_k}\mathcal{Q}(p^{1}_{m},y^{1}_{m},p^{0}_{m},y^{0}_{m}). \end{aligned} $$

(2.143)

Recall that any function $\mathcal {P}(p^1,y^1,p^0,y^0)$ or $\mathcal {Q}(p^1,y^1,p^0,y^0)$, that is invariant to changes in the units of measurement, can be written as a function of only 3M variables, namely the price relatives $p_{m}^1/p_{m}^0$ or the quantity relatives $y_{m}^1/y_{m}^0$, respectively, and the values $v_m^t$ (m = 1, …, M;t = 0, 1). This leads to further specifications of the indicators.

It is also useful to know that any indicator can be transformed into an index, and vice versa (Balk 2008, 128–129). In the process of transformation, however, certain properties may get lost.

2.1.2.1 The Main Formulas

The indicators materializing in this book will now be reviewed. The Laspeyres price indicator as function of prices and quantities is defined as

$$\displaystyle \begin{aligned} \mathcal{P}^{L}(p^1,y^1,p^0,y^0) \equiv (p^1 - p^0)\cdot y^0. \end{aligned} $$

(2.144)

This indicator can be written as a function of price relatives and (base period) values,

$$\displaystyle \begin{aligned} \mathcal{P}^{L}(p^1,y^1,p^0,y^0) = \sum_{m=1}^{M}(p_m^1/p_m^0 - 1)v_m^0. \end{aligned} $$

(2.145)

The Laspeyres quantity indicator is similarly defined as

$$\displaystyle \begin{aligned} \mathcal{Q}^{L}(p^1,y^1,p^0,y^0) \equiv (y^1 - y^0)\cdot p^0. \end{aligned} $$

(2.146)

Notice that $\mathcal {P}^{L}(p^1,y^1,p^0,y^0) + \mathcal {Q}^{L}(p^1,y^1,p^0,y^0) \neq p^1 \cdot y^1 - p^0 \cdot y^0$, except in special cases.

The Paasche price indicator, defined as

$$\displaystyle \begin{aligned} \mathcal{P}^{P}(p^1,y^1,p^0,y^0) \equiv (p^1 - p^0)\cdot y^1 \end{aligned} $$

(2.147)

can be written as a function of price relatives and (comparison period) values,

$$\displaystyle \begin{aligned} \mathcal{P}^{P}(p^1,y^1,p^0,y^0) = \sum_{m=1}^{M}(1 - p_m^0/p_m^1)v_m^1. \end{aligned} $$

(2.148)

The Paasche quantity indicator is defined as

$$\displaystyle \begin{aligned} \mathcal{Q}^{P}(p^1,y^1,p^0,y^0) \equiv (y^1 - y^0)\cdot p^1. \end{aligned} $$

(2.149)

Notice that $\mathcal {P}^{P}(p^1,y^1,p^0,y^0) + \mathcal {Q}^{P}(p^1,y^1,p^0,y^0) \neq p^1 \cdot y^1 - p^0 \cdot y^0$, except in special cases. However, $\mathcal {P}^{L}(p^1,y^1,p^0,y^0) + \mathcal {Q}^{P}(p^1,y^1,p^0,y^0) = \mathcal {P}^{P}(p^1,y^1,p^0,y^0) + \mathcal {Q}^{L}(p^1,y^1,p^0,y^0) = p^1 \cdot y^1 - p^0 \cdot y^0$.

The Bennet price indicator is usually defined by

$$\displaystyle \begin{aligned} \mathcal{P}^{B}(p^1,y^1,p^0,y^0) \equiv (1/2)(p^1 - p^0)\cdot(y^0 + y^1). \end{aligned} $$

(2.150)

It is the arithmetic mean of the Laspeyres and Paasche indicators; that is,

$$\displaystyle \begin{aligned} \mathcal{P}^{B}(p^1,y^1,p^0,y^0) = (1/2)\left[\mathcal{P}^{L}(p^1,y^1,p^0,y^0) + \mathcal{P}^{P}(p^1,y^1,p^0,y^0)\right], \end{aligned} $$

(2.151)

which can be written as

$$\displaystyle \begin{aligned} \mathcal{P}^{B}(p^1,y^1,p^0,y^0) = (1/2)\left[\sum_{m=1}^{M}(p_m^1/p_m^0 - 1)v_m^0 + \sum_{m=1}^{M}(1 - p_m^0/p_m^1)v_m^1\right]. \end{aligned} $$

(2.152)

The Bennet quantity indicator is defined by

$$\displaystyle \begin{aligned} \mathcal{Q}^{B}(p^1,y^1,p^0,y^0) \equiv (1/2)(y^1 - y^0)\cdot(p^0 + p^1). \end{aligned} $$

(2.153)

One easily verifies that $\mathcal {P}^{B}(p^1,y^1,p^0,y^0) + \mathcal {Q}^{B}(p^1,y^1,p^0,y^0) = p^1 \cdot y^1 - p^0 \cdot y^0$.

In the Bennet price indicator the price differences $p_m^1 - p_m^0$ are multiplied by arithmetic means of base and comparison period quantities $(y_m^0 + y_m^1)/2$. The Montgomery price indicator uses instead logarithmic mean values divided by logarithmic mean prices; that is,

$$\displaystyle \begin{aligned} \mathcal{P}^{M}(p^1,y^1,p^0,y^0) \equiv \sum_{m=1}^{M}\frac{\mathit{LM}(v_m^0,v_m^1)} {\mathit{LM}(p_m^0,p_m^1)}(p_m^1 - p_m^0). \end{aligned} $$

(2.154)

The Montgomery quantity indicator is defined as

$$\displaystyle \begin{aligned} \mathcal{Q}^{M}(p^1,y^1,p^0,y^0) \equiv \sum_{m=1}^{M}\frac{\mathit{LM}(v_m^0,v_m^1)} {\mathit{LM}(y_m^0,y_m^1)}(y_m^1 - y_m^0). \end{aligned} $$

(2.155)

Using the definition of the logarithmic mean one easily verifies that $\mathcal {P}^{M}(p^1,y^1,$ $p^0,y^0) + \mathcal {Q}^{M}(p^1,y^1,p^0,y^0) = p^1 \cdot y^1 - p^0 \cdot y^0$.

Appendix B: Decompositions of the Value-Added Ratio

Value added is defined as revenue minus the cost of intermediate inputs. Let the revenue ratio R ¹∕R ⁰ as in expression (2.14) be decomposed as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{R^1}{R^0} & = &\displaystyle P(p^1,y^1,p^0,y^0)Q(p^1,y^1,p^0,y^0) \\ & \equiv &\displaystyle P_R(1,0)Q_R(1,0), \end{array} \end{aligned} $$

(2.156)

for certain price and quantity indices, and let the intermediate inputs cost ratio $C_{\mathit {EMS}}^1/C_{\mathit {EMS}}^0$ be decomposed by, not neccessarily the same, price and quantity indices as

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{C_{\mathit{EMS}}^1}{C_{\mathit{EMS}}^0} & = &\displaystyle P(w_{\mathit{EMS}}^1,x_{\mathit{EMS}}^1,w_{\mathit{EMS}}^0, x_{\mathit{EMS}}^0)Q(w_{\mathit{EMS}}^1,x_{\mathit{EMS}}^1, w_{\mathit{EMS}}^0,x_{\mathit{EMS}}^0) \\ & \equiv &\displaystyle P_{\mathit{EMS}}(1,0)Q_{\mathit{EMS}}(1,0). \end{array} \end{aligned} $$

(2.157)

How can the (nominal) value added ratio

$$\displaystyle \begin{aligned} \frac{\mathit{VA}^1}{\mathit{VA}^0} = \frac{R^1 - C_{\mathit{EMS}}^1}{R^0 - C_{\mathit{EMS}}^0}\end{aligned} $$

then be decomposed into meaningful price and quantity components? It is clear that any of these components potentially requires a lot of data, namely $(w_{\mathit {EMS}}^1,x_{\mathit {EMS}}^1,w_{\mathit {EMS}}^0, x_{\mathit {EMS}}^0,p^1,y^1,p^0,y^0)$, data that are not always immediately available. Therefore, traditionally, one had recourse to so-called single deflation, where, for instance, a revenue-based price index is used to deflate nominal value added. That is, one sets $P_{\mathit {VA}}^{sd}(1,0) \equiv P_R(1,0)$, and let the quantity index be the remainder,

$$\displaystyle \begin{aligned} Q_{\mathit{VA}}^{sd}(1,0) \equiv \frac{\mathit{VA}^1}{\mathit{VA}^0} \Big/ P_R(1,0).{}\end{aligned} $$

(2.158)

Using the various definitions, this can be written as

$$\displaystyle \begin{aligned} \begin{array}{rcl} Q_{\mathit{VA}}^{sd}(1,0) & = &\displaystyle \frac{R^0}{\mathit{VA}^0}\frac{R^1/R^0}{P_R(1,0)} - \frac{C_{\mathit{EMS}}^0}{\mathit{VA}^0}\frac{C_{\mathit{EMS}}^1/C_{\mathit{EMS}}^0}{P_R(1,0)} {}\\ & = &\displaystyle \frac{R^0}{\mathit{VA}^0}Q_R(1,0) - \frac{C_{\mathit{EMS}}^0}{\mathit{VA}^0}Q_{\mathit{EMS}}(1,0)\frac{P_{\mathit{EMS}}(1,0)}{P_R(1,0)}, \end{array} \end{aligned} $$

(2.159)

but also as

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {Q_{\mathit{VA}}^{sd}(1,0) = } \\ & &\displaystyle \qquad \left[\frac{R^1}{\mathit{VA}^1}Q_R(1,0)^{-1} - \frac{C_{\mathit{EMS}}^1}{\mathit{VA}^1}\left(Q_{\mathit{EMS}}(1,0)\frac{P_{\mathit{EMS}}(1,0)}{P_R(1,0)}\right)^{-1}\right]^{-1}.{} \end{array} \end{aligned} $$

(2.160)

Two important observations must be made. First, if VA ^t > 0 (t = 0, 1) then $Q_{\mathit {VA}}^{sd}(1,0) > 0$. Second, it appears that in this setup the quantity component of the value added ratio depends on the price index of intermediate inputs relative to the price index of gross output, P _EMS(1, 0)∕P _R(1, 0). Since it is not at all certain that this ratio dwells in the neighbourhood of 1, this can create undesirable outcomes.

A variant is mentioned by SNA (2008, par. 15.135), namely $Q_{\mathit {VA}}^{sd'}(1,0) \equiv Q_R(1,0)$; that is, the growth rate of real value added is set equal to the growth rate of gross output. It appears that

$$\displaystyle \begin{aligned} Q_{\mathit{VA}}^{sd'}(1,0) = \frac{R^1/\mathit{VA}^1}{R^0/\mathit{VA}^0}Q_{\mathit{VA}}^{sd}(1,0). \end{aligned} $$

(2.161)

Thus, if the ratio of revenue to value added does not change then the two single-deflation variants coincide. In the literature this ratio is known as the Domar factor.

When the necessary data are available, SNA (2008, par. 15.133) prefers double deflation, which means that the value-added based quantity index is defined as³⁹

$$\displaystyle \begin{aligned} Q_{\mathit{VA}}^{dd1}(1,0) \equiv \frac{R^0}{\mathit{VA}^0}Q_R(1,0) - \frac{C_{\mathit{EMS}}^0}{\mathit{VA}^0}Q_{\mathit{EMS}}(1,0), \end{aligned} $$

(2.162)

or alternatively as

$$\displaystyle \begin{aligned} Q_{\mathit{VA}}^{dd2}(1,0) \equiv \left[\frac{R^1}{\mathit{VA}^1}Q_R(1,0)^{-1} - \frac{C_{\mathit{EMS}}^1}{\mathit{VA}^1}Q_{\mathit{EMS}}(1,0)^{-1}\right]^{-1}. \end{aligned} $$

(2.163)

Both quantity indices have a two-stage structure. The first stage is given by the quantity indices for gross output and intermediate inputs, Q _R(1, 0) and Q _EMS(1, 0), respectively.⁴⁰ The second stage is in the case of expression (2.162) a Laspeyres index (with weights from the base period), and in the case of expression (2.163) a Paasche index (with weights from the comparison period). Notice the minus signs for the parts concerning intermediate inputs.

It is then rather natural to propose the geometric mean of the two quantity indices as the desired quantity index. Put otherwise, the quantity component of the value added ratio is defined as the Fisher index of the subindices Q _R(1, 0) and Q _EMS(1, 0); that is,

$$\displaystyle \begin{aligned} Q_{\mathit{VA}}^F(1,0) \equiv \left[\dfrac{\dfrac{R^0}{\mathit{VA}^0}Q_R(1,0) - \dfrac{C_{\mathit{EMS}}^0}{\mathit{VA}^0}Q_{\mathit{EMS}}(1,0)} {\dfrac{R^1}{\mathit{VA}^1}(Q_R(1,0))^{-1} - \dfrac{C_{\mathit{EMS}}^1}{\mathit{VA}^1}(Q_{\mathit{EMS}}(1,0))^{-1}}\right]^{1/2}. \end{aligned} $$

(2.164)

Similarly, the price component of the value added ratio is defined as the Fisher index of the subindices P _R(1, 0) and P _EMS(1, 0); that is,

$$\displaystyle \begin{aligned} P_{\mathit{VA}}^F(1,0) \equiv \left[\dfrac{\dfrac{R^0}{\mathit{VA}^0}P_R(1,0) - \dfrac{C_{\mathit{EMS}}^0}{\mathit{VA}^0}P_{\mathit{EMS}}(1,0)} {\dfrac{R^1}{\mathit{VA}^1}(P_R(1,0))^{-1} - \dfrac{C_{\mathit{EMS}}^1}{\mathit{VA}^1}(P_{\mathit{EMS}}(1,0))^{-1}}\right]^{1/2}. \end{aligned} $$

(2.165)

One easily checks that $P_{\mathit {VA}}^F(1,0)Q_{\mathit {VA}}^F(1,0) = \mathit {VA}^1/\mathit {VA}^0$. This pair of indices was proposed by Geary (1944), though Karmel (1954) discloses some earlier sources.

For studying the behaviour of the quantity index $Q_{\mathit {VA}}^F(1,0)$ the following relation is useful:

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\frac{Q_{\mathit{VA}}^F(1,0)}{Q_R(1,0)} =} \\ & &\displaystyle \left[\dfrac{1 - \dfrac{C_{\mathit{EMS}}^0}{R^0}\dfrac{Q_{\mathit{EMS}}(1,0)}{Q_R(1,0)}}{1 - \dfrac{C_{\mathit{EMS}}^0}{R^0}} \dfrac{1 - \dfrac{C_{\mathit{EMS}}^1}{R^1}}{1 - \dfrac{C_{\mathit{EMS}}^1}{R^1}\left(\dfrac{Q_{\mathit{EMS}}(1,0)}{Q_R(1,0)}\right)^{-1}}\right]^{1/2}.{} \end{array} \end{aligned} $$

(2.166)

When VA ^t > 0 then $1 - C_{\mathit {EMS}}^t/R^t > 0$ (t = 0, 1). However, the function $Q_{VA}^F(1,0)$ is undefined when $Q_R(1,0) \leq (C_{\mathit {EMS}}^0/R^0)Q_{\mathit {EMS}}(1,0)$ or $Q_R(1,0) \geq (R^1/C_{\mathit {EMS}}^1)Q_{\mathit {EMS}}(1,0)$. Moreover, Eq. (2.166) implies the following relations:

$$\displaystyle \begin{aligned} \begin{array}{rcl} Q_R(1,0) > Q_{\mathit{EMS}}(1,0) & \Rightarrow &\displaystyle Q_{\mathit{VA}}^F(1,0) > Q_R(1,0) \\ Q_R(1,0) = Q_{\mathit{EMS}}(1,0) & \Rightarrow &\displaystyle Q_{\mathit{VA}}^F(1,0) = Q_R(1,0) \\ Q_R(1,0) < Q_{\mathit{EMS}}(1,0) & \Rightarrow &\displaystyle Q_{\mathit{VA}}^F(1,0) < Q_R(1,0) . \end{array} \end{aligned} $$

(2.167)

Karmel (1954), however, showed that in the case of chained indices these relations might be violated. Thus there may occur situations where Fisher indices are undefined.

An alternative decomposition, which does not exhibit this defect, can be developed as follows. For the logarithm of the value added ratio we get by repeated application of the logarithmic mean LM(a, b),

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\ln\left(\frac{\mathit{VA}^1}{\mathit{VA}^0}\right) = \frac{\mathit{VA}^1 - \mathit{VA}^0}{\mathit{LM}(\mathit{VA}^1,\mathit{VA}^0)}=}\\ & &\displaystyle \qquad \frac{R^1 - R^0}{\mathit{LM}(\mathit{VA}^1,\mathit{VA}^0)} - \frac{C_{\mathit{EMS}}^1 - C_{\mathit{EMS}}^0}{\mathit{LM}(\mathit{VA}^1,\mathit{VA}^0)}= \\ & &\displaystyle \qquad \frac{L(R^1,R^0)\ln(R^1/R^0)}{\mathit{LM}(\mathit{VA}^1,\mathit{VA}^0)} - \frac{LM(C_{\mathit{EMS}}^1,C_{\mathit{EMS}}^0)\ln(C_{\mathit{EMS}}^1/C_{\mathit{EMS}}^0)}{\mathit{LM}(\mathit{VA}^1,\mathit{VA}^0)}. \end{array} \end{aligned} $$

(2.168)

Using the decompositions of the revenue ratio and the intermediate inputs cost ratio, the logarithm of the value added ratio can be expressed as

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\ln\left(\frac{\mathit{VA}^1}{\mathit{VA}^0}\right) = }\\ & &\displaystyle \qquad \frac{\mathit{LM}(R^1,R^0)\ln(P_R(1,0)Q_R(1,0))} {\mathit{LM}(\mathit{VA}^1,\mathit{VA}^0)} - \\ & &\displaystyle \qquad \frac{\mathit{LM}(C_{\mathit{EMS}}^1,C_{\mathit{EMS}}^0)\ln(P_{\mathit{EMS}}(1,0)Q_{\mathit{EMS}}(1,0))} {\mathit{LM}(\mathit{VA}^1,\mathit{VA}^0)}. \end{array} \end{aligned} $$

(2.169)

This can be rearranged as

$$\displaystyle \begin{aligned} \frac{\mathit{VA}^1}{\mathit{VA}^0} = \frac{P_R(1,0)^{\phi(1,0)}}{P_{\mathit{EMS}}(1,0))^{\psi(1,0)}} \frac{Q_R(1,0)^{\phi(1,0)}}{Q_{\mathit{EMS}}(1,0))^{\psi(1,0)}}, \end{aligned} $$

(2.170)

where ϕ(1, 0) ≡LM(R ¹, R ⁰)∕LM(V A ¹, V A ⁰), that is, mean revenue over mean value added, and $\psi (1,0) \equiv \mathit {LM}(C_{\mathit {EMS}}^1,C_{\mathit {EMS}}^0)/\mathit {LM}(VA^1,VA^0)$, that is, mean intermediate inputs cost over mean value added. Thus, value-added based price and quantity indices can rather naturally be defined by

$$\displaystyle \begin{aligned} \begin{array}{rcl} P_{\mathit{VA}}^{MV}(1,0) & \equiv &\displaystyle \frac{P_R(1,0)^{\phi(1,0)}}{P_{\mathit{EMS}}(1,0))^{\psi(1,0)}} \end{array} \end{aligned} $$

(2.171)

$$\displaystyle \begin{aligned} \begin{array}{rcl} Q_{\mathit{VA}}^{MV}(1,0) & \equiv &\displaystyle \frac{Q_R(1,0)^{\phi(1,0)}}{Q_{\mathit{EMS}}(1,0))^{\psi(1,0)}}. \end{array} \end{aligned} $$

(2.172)

These indices generalize the conventional Montgomery-Vartia indices (Appendix A). Their disadvantage is that P _R(1, 0) = P _EMS(1, 0) (or Q _R(1, 0) = Q _EMS(1, 0)) not necessarily implies that $P_{\mathit {VA}}^{MV}(1,0) = P_R(1,0)$ (or $Q_{\mathit {VA}}^{MV}(1,0) = Q_R(1,0)$). Formally stated, these indices fail the Equality Test. The reason behind this failure is that generally

$$\displaystyle \begin{aligned} \phi_{R}(1,0) - \phi_{\mathit{EMS}}(1,0) = \frac{\mathit{LM}(R^1,R^0) - \mathit{LM}(C_{\mathit{EMS}}^1,C_{\mathit{EMS}}^0)}{\mathit{LM}(\mathit{VA}^1,\mathit{VA}^0)} \end{aligned} $$

(2.173)

is close but not equal to 1 because LM(a, 1) is a concave function. Fortunately, in ‘normal’ situations the discrepancy appears to be unimportant.

Summarizing the foregoing, we see that

• single deflation is simple, leads to a quantity index that is always positive, but this index may be biased;

• double deflation by a Fisher index leads to a quantity index that is unbiased but may not always be well-defined;

• double deflation by a Montgomery-Vartia index leads to a quantity index that is always well-defined but fails the Equality Test.

Thus, there is no theoretically entirely satisfactory solution to the problem of decomposing a value-added ratio into a price index and a quantity index.

Appendix C: The Domar Factor

2.1.1 And the TFP Index

According to the definition of the gross-output based TFP index⁴¹

$$\displaystyle \begin{aligned} \ln \mathit{ITFPROD}_Y(1,0) = \ln Q_R(1,0) - \ln Q_C(1,0). \end{aligned} $$

(2.174)

We know that $C^t = C^t_{\mathit {KL}} + C^t_{\mathit {EMS}}$ (t = 0, 1). It is assumed that the primary inputs cost ratio is decomposed into a price and a quantity index, $C^1_{\mathit {KL}}/C^0_{\mathit {KL}} = P_{\mathit {KL}}(1,0)Q_{\mathit {KL}}(1,0)$. It is also assumed that the intermediate inputs cost ratio is decomposed into a price and a quantity index, $C^1_{\mathit {EMS}}/C^0_{\mathit {EMS}} = P_{\mathit {EMS}}(1,0) \times Q_{\mathit {EMS}}(1,0)$. Then Q _C(1, 0) can be defined as the Montgomery-Vartia index of Q _KL(1, 0) and Q _EMS(1, 0);⁴² that is,

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\ln Q_{C}(1,0) = }\\ & &\displaystyle \quad \frac{\mathit{LM}(C_{\mathit{KL}}^{1},C_{\mathit{KL}}^{0})}{\mathit{LM}(C^{1},C^{0})}\ln Q_{\mathit{KL}}(1,0) + \frac{\mathit{LM}(C_{\mathit{EMS}}^{1},C_{\mathit{EMS}}^{0})}{\mathit{LM}(C^{1},C^{0})}\ln Q_{\mathit{EMS}}(1,0), \end{array} \end{aligned} $$

(2.175)

where LM(a, b) is the logarithmic mean of a and b. Then, substituting this expression into expression (2.174), we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\ln \mathit{ITFPROD}_Y(1,0) = \ln Q_{R}(1,0)}\\ & &\displaystyle \quad - \dfrac{\mathit{LM}(C_{\mathit{KL}}^{1},C_{\mathit{KL}}^{0})}{\mathit{LM}(C^{1},C^{0})}\ln Q_{\mathit{KL}}(1,0) - \dfrac{\mathit{LM}(C_{\mathit{EMS}}^{1},C_{\mathit{EMS}}^{0})}{\mathit{LM}(C^{1},C^{0})}\ln Q_{\mathit{EMS}}(1,0).{} \end{array} \end{aligned} $$

(2.176)

Next, the definition of value added is rewritten as $R^t = \mathit {VA}^t + C^t_{\mathit {EMS}}$ (t = 0, 1). Let Q _R(1, 0) be defined as the Montgomery-Vartia index of Q _VA(1, 0) and Q _EMS(1, 0). Then the logarithm of the value-added based quantity index can be backed out as

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\ln Q_{VA}(1,0) = }\\ & &\displaystyle \qquad \frac{\mathit{LM}(R^{1},R^{0})}{\mathit{LM}(\mathit{VA}^{1},\mathit{VA}^{0})}\ln Q_{R}(1,0) - \frac{\mathit{LM}(C_{\mathit{EMS}}^{1},C_{\mathit{EMS}}^{0})}{\mathit{LM}(\mathit{VA}^{1},\mathit{VA}^{0})}\ln Q_{\mathit{EMS}}(1,0). {} \end{array} \end{aligned} $$

(2.177)

Substituting this into the definition of the value-added based TFP index delivers

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\ln \mathit{ITFPROD}_{\mathit{VA}}(1,0) = \frac{\mathit{LM}(R^{1},R^{0})}{\mathit{LM}(\mathit{VA}^{1},\mathit{VA}^{0})}\Big[\ln Q_{R}(1,0)} \\ & &\displaystyle \qquad - \frac{\mathit{LM}(\mathit{VA}^{1},\mathit{VA}^{0})}{\mathit{LM}(R^{1},R^{0})}\ln Q_{\mathit{KL}}(1,0) - \frac{\mathit{LM}(C_{\mathit{EMS}}^{1},C_{\mathit{EMS}}^{0})}{\mathit{LM}(R^{1},R^{0})}\ln Q_{\mathit{EMS}}(1,0)\Big].{} \end{array} \end{aligned} $$

(2.178)

Combining the two expressions (2.176) and (2.178) by eliminating $\ln Q_{R}(1,0)$ delivers

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle {\ln \mathit{ITFPROD}_{\mathit{VA}}(1,0) = \frac{\mathit{LM}(R^{1},R^{0})}{\mathit{LM}(\mathit{VA}^{1},\mathit{VA}^{0})} \Big[\ln \mathit{ITFPROD}_{Y}(1,0) } \\ & &\displaystyle \qquad + \: \left(\frac{\mathit{LM}(C_{\mathit{KL}}^{1},C_{\mathit{KL}}^{0})}{\mathit{LM}(C^{1},C^{0})} - \frac{\mathit{LM}(\mathit{VA}^{1},\mathit{VA}^{0})}{\mathit{LM}(R^{1},R^{0})}\right)\ln Q_{\mathit{KL}}(1,0) \\ & &\displaystyle \qquad + \: \left(\frac{\mathit{LM}(C_{\mathit{EMS}}^{1},C_{\mathit{EMS}}^{0})}{\mathit{LM}(C^{1},C^{0})} - \frac{\mathit{LM}(C_{\mathit{EMS}}^{1},C_{\mathit{EMS}}^{0})}{\mathit{LM}(R^{1},R^{0})}\right)\ln Q_{\mathit{EMS}}(1,0)\Big].{} \end{array} \end{aligned} $$

(2.179)

The factor in front of the square brackets,

$$\displaystyle \begin{aligned} D(1,0) \equiv \mathit{LM}(R^{1},R^{0})/\mathit{LM}(\mathit{VA}^{1},\mathit{VA}^{0}), \end{aligned} $$

(2.180)

is known as the Domar factor: the ratio of (mean) revenue over (mean) value added. One easily checks that if revenue R ^t is equal to cost C ^t (or value added V A ^t is equal to primary inputs cost $C_{\mathit {KL}}^{t}$, or profit Π^t is equal to 0) (t = 0, 1), then expression (2.179) reduces to

$$\displaystyle \begin{aligned} \ln \mathit{ITFPROD}_{\mathit{VA}}(1,0) = D(1,0)\ln \mathit{ITFPROD}_{Y}(1,0). \end{aligned} $$

(2.181)

This corresponds to Proposition 1 of Gollop (1979). See Jorgenson et al. (2005, 298) for a modern derivation under the usual neo-classical assumptions.

2.1.2 And the Labour Productivity Index

According to the definition of the value-added based labour productivity index, expression (2.81),

$$\displaystyle \begin{aligned} \ln \mathit{ILPROD}_{\mathit{VA}}(1,0) = \ln Q_{\mathit{VA}}(1,0) - \ln Q_{L}(1,0). \end{aligned} $$

(2.182)

Substituting expressions (2.177) and (2.180) delivers

$$\displaystyle \begin{aligned} \ln \mathit{ILPROD}_{\mathit{VA}}(1,0) = D(1,0)\ln Q_{R}(1,0) \end{aligned} $$

(2.183)

$$\displaystyle \begin{aligned}- \: \frac{\mathit{LM}(C_{\mathit{EMS}}^{1},C_{\mathit{EMS}}^{0})}{\mathit{LM}(\mathit{VA}^{1},\mathit{VA}^{0})}\ln Q_{\mathit{EMS}}(1,0) - \ln Q_{L}(1,0). \end{aligned}$$

According to the definition of the gross-output based labour productivity index, expression (2.61),

$$\displaystyle \begin{aligned} \ln \mathit{ILPROD}_{Y}(1,0) = \ln Q_{R}(1,0) - \ln Q_{L}(1,0). \end{aligned} $$

(2.184)

Substituting this into expression (2.183) delivers

$$\displaystyle \begin{aligned} \ln \mathit{ILPROD}_{\mathit{VA}}(1,0) = D(1,0)\ln \mathit{ILPROD}_{Y}(1,0) \end{aligned} $$

(2.185)

$$\displaystyle \begin{aligned}- \: \frac{\mathit{LM}(C_{\mathit{EMS}}^{1},C_{\mathit{EMS}}^{0})}{\mathit{LM}(\mathit{VA}^{1},\mathit{VA}^{0})}\ln Q_{\mathit{EMS}}(1,0) + \left(D(1,0) - 1\right)\ln Q_{L}(1,0). \end{aligned}$$

The assumption that Q _R(1, 0) is the Montgomery-Vartia index of Q _VA(1, 0) and Q _EMS(1, 0) implies that the sum of its weights is approximately equal to 1; that is,

$$\displaystyle \begin{aligned} \frac{\mathit{LM}(\mathit{VA}^{1},\mathit{VA}^{0})}{\mathit{LM}(R^{1},R^{0})} + \frac{\mathit{LM}(C_{\mathit{EMS}}^{1},C_{\mathit{EMS}}^{0})}{\mathit{LM}(R^{1},R^{0})} \approx 1. \end{aligned} $$

(2.186)

Employing the definition of the Domar factor, this implies that

$$\displaystyle \begin{aligned} \frac{\mathit{LM}(C_{\mathit{EMS}}^{1}, C_{\mathit{EMS}}^{0})}{\mathit{LM}(\mathit{VA}^{1},\mathit{VA}^{0})} \approx D(1,0) - 1. \end{aligned} $$

(2.187)

Substituting this into expression (2.185) delivers

$$\displaystyle \begin{aligned} \ln \mathit{ILPROD}_{\mathit{VA}}(1,0) \approx D(1,0)\ln \mathit{ILPROD}_{Y}(1,0) \end{aligned} $$

(2.188)

$$\displaystyle \begin{aligned} \qquad\quad+ \: \left(D(1,0) - 1\right)\ln \left(\frac{Q_{L}(1,0)}{Q_{\mathit{EMS}}(1,0)}\right), \end{aligned}$$

which is our final result. Notice that the second term at the right-hand side of this approximate equality does not vanish if revenue R ^t equals cost C ^t. The term vanishes if labour quantity change Q _L(1, 0) equals the aggregate quantity change of intermediate inputs Q _EMS(1, 0).