On the Relationship Between Gross-Output and Value-Added Based

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CENTRE FOR APPLIED ECONOMIC RESEARCH

WORKING PAPER (2003/05)

On the Relationship Between Gross-Output and Value-Added Based Productivity Measures: The Importance of the Domar Factor By Bert M. Balk

ISSN 13 29 12 70 ISBN 0 7334 2117 2 www.caer.unsw.edu.au

On the Relation Between Gross-Output and Value-Added Based Productivity Measures: The Importance of the Domar Factor Bert M. Balk∗ Erasmus Research Institute of Management Erasmus University Rotterdam and Methods and Informatics Department Statistics Netherlands P. O. Box 4000, 2270 JM Voorburg, The Netherlands E-mail [email protected] December 23, 2003

Abstract In this paper I consider the relation between gross-output and valueadded based total factor productivity (TFP) measures. It appears that, without requiring any (micro-)economic theory, a conditional relationship between TFP indexes can be derived, in which the Domar factor plays an important role. At the same time it turns out that gross-output and value-added based TFP indicators (difference type measures) always coincide. In the Divisia index framework and ∗

The views expressed in this paper are those of the author and do not necessarily reflect any policy of Statistics Netherlands. Earlier versions were presented at the 8th European Workshop on Efficiency and Productivity Analysis, Oviedo, 25-27 September 2003, and at the Economic Measurement Group meeting, Sydney, 11-12 December 2003.

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maintaining the classical assumptions (profit maximization and a production technology that exhibits globally constant returns to scale), it appears that both TFP indexes measure technological change, albeit in a dual way. In establishing this result, no separability assumptions are involved. Both indexes are in general path-dependent. Pathindependency of the gross-output based TFP index requires that the technology exhibits Hicks input neutrality, whereas path-independency of the value-added based TFP index requires Hicks value added neutrality. These two concepts of neutrality are, however, not dual. JEL Classification: C43, D24, O47. Keywords: Total Factor Productivity; Domar factor; gross output; value added; technological change; Hicks neutrality; path-(in)dependency.

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Introduction

Browsing through empirical studies on productivity, as presented at conferences and in journals, it appears that they can be classified into two groups according to the productivity measure used. The first group uses gross output based productivity measures, whereas the second group uses value added based measures. The first group essentially conceives any production unit (be it a firm, an industry, or an economy) as an input-output mechanism that uses capital, labour, energy, materials, and services to produce a small or large number of goods and/or services. The second group essentially conceives all intermediate inputs as negative outputs that can in some monetary way be netted with produced outputs. Put otherwise, any production unit is conceived as a mechanism that turns capital and labour into (real) value added. In microdata studies one easily encounters both views, whereas in meso- and macrodata studies the second view is dominating.1 There would be no problem when, applied to the same data, both productivity measures would yield the same outcome. This, however, is usually not the case and the issue becomes particularly pressing when it comes to interpretation of the two measures. It is well known that under standard economic assumptions the gross-output based productivity index measures 1 See the OECD (2001) productivity measurement manual for an extensive discussion of the two approaches.

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technological change. But what about the value-added based index? Empirical studies that use the value added output concept usually suggest that under appropriately modified standard assumptions the value-added based productivity index would also measure technological change. Is this suggestion warranted? Are both sets of assumptions consistent with each other? Are there, depending on the context, two measures of technological change? In a still unpublished note, Schreyer (2000) nicely summarized the available evidence:2 “1. Under standard assumptions (i.e. a technology exhibiting constant returns to scale and a profit maximizing unit) grossoutput based productivity growth is a path-independent measure of technological change if and only if technological change is Hicks-neutral with regard to all (primary and intermediate) inputs. Path-independence means that the productivity index between two points in time depends only on the realisations of input and output variables at these two points in time, and not on the specific ‘path’ that links the two observations. This is a re-statement of Hulten (1973). 2. Value-added based productivity measures can be defined within standard assumptions about production technology. In particular, separability of the gross production function in primary and intermediate inputs is not a necessary condition for the existence of a value-added based productivity measure. Gross-output based productivity change and value-added based productivity change are not identical measures of technological change. However, they are related by a simple relation, the share of current price valueadded in gross output. This has been shown by Bruno (1978). 3. For a given production function with Hicks-neutral technological change, value-added based productivity change is not, in general, a path-independent measure of technological change. This statement builds on Sato (1976).” His fourth point reads: 2

I have edited his text slightly to let its wording correspond to that used in the present paper.

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“4. Value-added based productivity growth will be a path-independent measure of technological change if the underlying production technology is linearly homogeneous and separable in a real valueadded function and intermediate inputs. The real value-added function combines primary inputs and technological change. Technological change is constrained to shifts of the real value-added function and operates independently of intermediate inputs.” The conditions stated here, however, turn out to be too restrictive, as will be shown in this paper.3 The ratio alluded to in Schreyer’s second point – the share of currentprice value added in the value of gross output – happens to be the inverse of the Domar (1961) factor. This factor will play an important role in this paper. In this paper I reconsider the relation between gross-output and valueadded based total factor productivity (TFP) measures, both in a discrete and a continuous time framework. With respect to indexes it appears that, without requiring any (micro-) economic theory, a relationship can be derived in which the Domar factor plays a crucial role. Instead of indexes, which are ratio type measures, one can opt for indicators, which are difference type measures.4 It then turns out that gross-output and value-added based TFP indicators always coincide. In the Divisia index framework and maintaining the classical assumptions (i.e. profit maximization and a production technology that exhibits globally constant returns to scale), it appears that both TFP indexes measure technological change, albeit in a dual way. In establishing this result, no separability assumptions are involved. Both indexes will in general be path-dependent. Path-independency of the gross-output based TFP index requires that the technology exhibits Hicks input neutrality, whereas path-independency of the value-added based TFP index requires Hicks value added neutrality. These two concepts of neutrality are, however, not dual. The lay-out of this paper is as follows. Section 2 defines model and notation. Sections 3 and 4 consider, in discrete time, ratio and difference measures respectively. The main contribution here is Theorem 1. Section 5 is devoted to the continuous time analogs of the measures discussed in the 3

An important difference with Schreyer’s review is that I will treat the multiple output case explicitly. 4 See Diewert (1998) for a general introduction into this topic.

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previous sections. Section 6 then turns to the interpretation of the two TFP indexes. In a detailed way it is shown which assumptions at what place are needed to reduce these indexes to measures of technological change. Also the issue of path-independency will be discussed. Section 7 concludes by pointing out some practical uses that can be made of the insights obtained in this paper.

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Model and notation

Let us consider a single production unit. This could be an establishment, a firm, an industry, or even an entire economy. At the output side of this unit t there are M commodities, each with their price ptm and quantity ym , where m = 1, ..., M , and t denotes an accounting period. Similarly, at the input side there are N commodities, each with their price wnt and quantity xtn , where n = 1, ..., N . Where possible, vector notation will be used. All prices are assumed to be positive and all quantities are assumed to be non-negative. The revenue, that is, the nominal value of the production unit’s gross output, during the accounting period t is r t ≡ pt · y t =

M X

t ptm ym ,

(1)

wnt xtn .

(2)

m=1

whereas the production cost is given by ct ≡ w t · xt =

N X n=1

The unit’s profit is given by rt − ct , whereas the ratio rt /ct is called profitability. To define value added, additional notation must be introduced. All the inputs are assumed to be allocatable to three, mutually disjunct, categories, namely capital (K), labour (L), and intermediate inputs (M). The entire input t t , wM ) = price and quantity vectors can then be partitioned as wt = (wKL t t t t t t t t t (wK , wL , wM ) and x = (xKL , xM ) = (xK , xL , xM ) respectively. Total cost ct t t can be decomposed as ct = ctKL + ctM = wKL · xtKL + wM · xtM . Value added (VA) is defined as revenue minus the cost of intermediate inputs, that is

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t V At ≡ rt − ctM = pt · y t − wM · xtM ,

(3)

which is assumed to be positive. Capital and labour are called primary inputs. Intermediate inputs, usually further subdivided into energy, materials and services, are acquired from other production units or imported.5 The value added concept basically conceives the production unit as producing money from the primary input categories capital and labour.

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Ratio type measures

The comparison is focussed at two periods: a base period t = 0 and a comparison period t = 1. These periods may be adjacent to each other, or far apart. Following Balk (2003), the gross-output-based TFP index will be generically defined as the real component of the ratio of comparison period profitability, (r1 /c1 ), over base period profitability, (r0 /c0 ). It is natural to start with a decomposition of this ratio, which can also be written as (r1 /r0 )/(c1 /c0 ), into components. It is assumed that the revenue ratio can be decomposed into two parts, r1 /r0 = Po (p1 , y 1 , p0 , y 0 )Qo (p1 , y 1 , p0 , y 0 ),

(4)

where Po (.) is a price index and Qo (.) is a quantity index, both operating on the output prices and quantities of both periods. Likewise, it is assumed that the cost ratio can be decomposed as c1 /c0 = Pi (w1 , x1 , w0 , x0 )Qi (w1 , x1 , w0 , x0 ),

(5)

where Pi (.) is a price index and Qi (.) is a quantity index, both operating on the input prices and quantities of both periods. The gross-output based index of TFP is then defined as the ratio of the output quantity index and the input quantity index, IGOT F P ≡ Qo (p1 , y 1 , p0 , y 0 )/Qi (w1 , x1 , w0 , x0 ), 5

(6)

If the production unit considered is an entire economy then all intermediate inputs are imported. It is thereby assumed that all inter-firm or inter-industry flows cancel, which is the case when input prices are equal to output prices (i.e. there is no tax wedge). See Gollop (1987).

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or, using logarithms, ln IGOT F P = ln Qo (p1 , y 1 , p0 , y 0 ) − ln Qi (w1 , x1 , w0 , x0 ).

(7)

In order to compare this index with a value-added based index of TFP, it is useful to consider an other decomposition of the cost ratio c1 /c0 . We employ thereby the simple but powerful tool called the logarithmic mean. This mean is, for two positive real numbers a and b, defined by L(a, b) ≡ (a − b)/ ln(a/b) and L(a, a) ≡ a. It is easy to check that the function L(.) has all of the properties one expects a symmetric mean to exhibit. The logarithmic mean allows one to switch between a difference and a ratio.6 Using the definition of the logarithmic mean repeatedly, one obtains ln(c1 /c0 ) c1 − c0 = L(c1 , c0 ) c1 − c0 c1 − c0 = KL 1 0KL + M 1 0M L(c , c ) L(c , c ) 1 0 L(cKL , cKL ) ln(c1KL /c0KL ) L(c1M , c0M ) ln(c1M /c0M ) = + . L(c1 , c0 ) L(c1 , c0 )

(8)

It is assumed that the primary input cost ratio can be decomposed as 1 0 1 0 c1KL /c0KL = Pi0 (wKL , x1KL , wKL , x0KL )Q0i (wKL , x1KL , wKL , x0KL ),

(9)

and that the intermediate input cost ratio can be decomposed as 1 0 1 0 c1M /c0M = Pi00 (wM , x1M , wM , x0M )Q00i (wM , x1M , wM , x0M ),

(10)

where Pi0 (.) and Pi00 (.) are input price indexes, and Q0i (.) and Q00i (.) are input quantity indexes. Note that the functional forms of the price indexes Pi (.), 6

The logarithmic mean was introduced in the economics literature by T¨ornqvist in 1935 in an unpublished memo of the Bank of Finland; see T¨ornqvist, Vartia and Vartia (1985). It has the following properties: (1) min(a, b) ≤ L(a, b) ≤ max(a, b); (2) L(a, b) is continuous; (3) L(λa, λb) = λL(a, b) (λ > 0); (4) L(a, b) = L(b, a). A simple proof of the fact that (ab)1/2 ≤ L(a, b) ≤ (a + b)/2 was provided by Lorenzen (1990).

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Pi0 (.) and Pi00 (.), and those of the quantity indexes Qi (.), Q0i (.) and Q00i (.), may or may not be the same. Substituting (9) and (10) into (8), one obtains ln(c1 /c0 ) = L(c1KL , c0KL ) ln Pi0 (.) L(c1M , c0M ) ln Pi00 (.) + L(c1 , c0 ) L(c1 , c0 ) L(c1KL , c0KL ) ln Q0i (.) L(c1M , c0M ) ln Q00i (.) + + . L(c1 , c0 ) L(c1 , c0 )

(11)

An alternative gross-output based index of TFP may now be defined as ln IGOT F P ∗ ≡ ln Qo (p1 , y 1 , p0 , y 0 ) − ln Q∗i (w1 , x1 , w0 , x0 )

(12)

ln Q∗i (w1 , x1 , w0 , x0 ) ≡ L(c1KL , c0KL ) 1 0 ln Q0i (wKL , x1KL , wKL , x0KL ) 1 0 L(c , c ) L(c1M , c0M ) 1 0 ln Q00i (wM , x1M , wM + , x0M ). L(c1 , c0 )

(13)

where

Thus, instead of the single aggregate input quantity index in (7), we have obtained an index consisting of two parts, the first corresponding to primary inputs and the second to intermediate inputs. Put otherwise, Q∗i (w1 , x1 , w0 , x0 ) 1 0 is a two-stage index: in the first stage Q0i (wKL , x1KL , wKL , x0KL ) aggregates the 00 1 1 0 0 primary input quantities, whereas Qi (wM , xM , wM , xM ) aggregates the intermediate input quantities; in the second stage these two subindexes are aggregated by means of a Montgomery (1929), (1937) - Vartia-I (1974), (1976) index. Notice that Q∗i (w1 , x1 , w0 , x0 ) is not a weighted mean of these subindexes, since the weights do not add up to 1, L(c1KL , c0KL ) L(c1M , c0M ) + ≤ 1. L(c1 , c0 ) L(c1 , c0 )

(14)

This inequality follows immediately from the fact that L(a, 1) is a concave function of a, since this implies that 8

 1  1 0 1 0 L(c , c ) + L(c , c ) KL KL M M c0 c0M c0 1 0 L(c /c , 1) + L(c1M /c0M , 1) = KL KL KL c0 c0 c0 c1KL c0M c1M + 0 0 , 1) ≤ L( KL c0 c0KL c cM 1 0 = L(c /c , 1),

(15)

where use was made of the linear homogeneity (property (3)) of the logarithmic mean. In general the two-stage index Q∗i (w1 , x1 , w0 , x0 ) will differ from the singlestage index Qi (w1 , x1 , w0 , x0 ). If the functional form of the indices Qi (.), Q0i (.) and Q00i (.) were to be the Montgomery-Vartia form, the two-stage index would coincide with the single-stage index, since the Montgomery-Vartia index is consistent in aggregation (see Balk 1995). As Diewert (1978a) has shown, the Montgomery-Vartia index differentially approximates to the second order any superlative index, provided that the price and quantity changes are small between the two periods. This in turn implies that if the indices Qi (.), Q0i (.) and Q00i (.) are superlative (but not necessarily of the same form), then Q∗i (w1 , x1 , w0 , x0 ) will also differentially approximate Qi (w1 , x1 , w0 , x0 ). The value-added based TFP index is generically defined as the real component of the value added to primary input cost ratio, which is given by (V A1 /c1KL )/(V A0 /c0KL ) = (V A1 /V A0 )/(c1KL /c0KL ). Decomposing the ratio V A1 /V A0 into components yields ln(V A1 /V A0 ) V A1 − V A0 = L(V A1 , V A0 ) r1 − r0 c1M − c0M = − L(V A1 , V A0 ) L(V A1 , V A0 ) L(r1 , r0 ) ln(r1 /r0 ) L(c1M , c0M ) ln(c1M /c0M ) = − . L(V A1 , V A0 ) L(V A1 , V A0 )

(16)

Using (4), (10), and (9), a value-added based index of TFP can then be defined as 9

L(r1 , r0 ) ln IV AT F P ≡ ln Qo (p1 , y 1 , p0 , y 0 ) − 1 0 L(V A , V A )

(17)

L(c1M , c0M ) 1 0 1 0 ln Q00i (wM , x1M , wM , x0M ) − ln Q0i (wKL , x1KL , wKL , x0KL ), 1 0 L(V A , V A ) which can be rearranged as L(r1 , r0 ) ln Qo (p1 , y 1 , p0 , y 0 ) 1 0 L(V A , V A ) 1 0 L(cM , cM ) 1 0 − ln Q00i (wM , x1M , wM , x0M ) 1 0 L(r , r )  L(V A1 , V A0 ) 0 1 1 0 0 ln Qi (wKL , xKL , wKL , xKL ) . − L(r1 , r0 ) 

ln IV AT F P =

(18)

The part between brackets has the same structure as ln IGOT F P ∗ , except that the weights of the intermediate inputs part and the primary inputs part are different. The following result is immediate: Theorem 1 For all prices and quantities, ln IV AT F P =

L(r1 , r0 ) ln IGOT F P ∗ L(V A1 , V A0 )

(19)

if and only if rt = ct (t = 0, 1). Proof: Note that (19) holds if and only if L(c1M , c0M ) L(c1M , c0M ) 1 0 − ln Q00i (wM , x1M , wM , x0M ) = L(r1 , r0 ) L(c1 , c0 ) !

L(c1KL , c0KL ) L(V A1 , V A0 ) 1 0 − ln Q0i (wKL , x1KL , wKL , x0KL ). L(c1 , c0 ) L(r1 , r0 ) !

The proof of the ”if”-part rests on the fact that rt = ct is equivalent to ctKL = V At , which follows immediately from the definitions. For the ”only if”-part, we note that the intermediate inputs quantity 1 0 index, Q00i (wM , x1M , wM , x0M ), is independent of the primary inputs quantity 0 1 1 0 index, Qi (wKL , xKL , wKL , x0KL ). Thus the identity can only hold for all values 10

of the variables if both expressions between brackets are identically equal to 0. This implies in particular that L(r1 , r0 ) = L(c1 , c0 ) for all r1 , r0 , c1 , c0 . Since the logarithmic mean is linearly homogeneous, this equation can be written as (r0 /c0 )L(r1 /r0 , 1) = L(c1 /c0 , 1). Since the three variables r0 /c0 , r1 /r0 , and c1 /c0 are obviously independent, it must follow that r0 /c0 = 1 and r1 /r0 = c1 /c0 . QED A couple of remarks are worth making. First, the ratio of average revenue to average value added, L(r1 , r0 )/L(V A1 , V A0 ), is greater than 1.7 Thus, provided that revenue equals cost, value-added based TFP change will be greater than gross-output based TFP change. Second, the condition that revenue equals cost, rt = ct or ctKL = V At , means that ln(r1 /r0 )−ln(c1 /c0 ) = 0 as well as ln(V A1 /V A0 )−ln(c1KL /c0KL ) = 0. This in turn implies that both TFP indexes can be expressed in dual form, that is ln IGOT F P ∗ = − ln Po (p1 , y 1 , p0 , y 0 ) + L(c1KL , c0KL ) L(c1 , c0 )

1 0 ln Pi0 (wKL , x1KL , wKL , x0KL ) +

L(c1M , c0M ) L(c1 , c0 )

(20)

1 0 ln Pi00 (wM , x1M , wM , x0M ).

and ln IV AT F P ≡ −

L(r1 , r0 ) ln Po (p1 , y 1 , p0 , y 0 ) + L(V A1 , V A0 )

(21)

L(c1M , c0M ) 1 0 1 0 ln Pi00 (wM , x1M , wM , x0M ) + ln Pi0 (wKL , x1KL , wKL , x0KL ). L(V A1 , V A0 ) respectively. Third, although in general revenue will be unequal to cost, their difference – which is profit – will often be relatively small, so that expression (19) will hold approximately. Fourth, the result stated in Theorem 1 does not require any (micro-) economic theory. It is interesting to observe that in a Cobb-Douglas production function context expression (19) basically, although not rigourously, was derived by Domar (1961); see his equation (4.6). Therefore the ratio of revenue 7

Unless cM = 0, which corresponds to the “closed economy” case of Gollop (1987).

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over value added is called the Domar factor. It is interesting to quote his evaluation of the two measures: “Of the two, A [i.e. gross-output based TFP] seems to have a clearer meaning: it is derived from a production equation containing all identifiable outputs and inputs, without any arbitrary exclusions from either side. But this Residual is too absolute, so to speak; a small A may be simply caused by the “thinness” of the industry, that is by the small degree of transformation its raw materials undergo. The Residual A0 [i.e. value-added based TFP] is free from this defect, and its use greatly simplifies aggregation and integration of industries, but the meaning of a Residual arising, say, from the production of shoes without leather is less easy to understand. Perhaps it is best to think of A0 as an A adjusted in a way for the “thickness” of the industry rather than as a Residual directly obtained from a peculiar value added index, though one’s attitude may be merely a matter of convenience and habit.” (p. 726) Put otherwise, in general both measures of TFP change convey a different message. It remains to be seen whether this is an artifact of the use of ratio-type measures. This is the topic of the next section. Before turning to this section, however, I will briefly discuss the relationship between the Domar factor and so-called Domar aggregation. Consider an ensemble of K production units. Under the value-added output concept, these units can be considered as disjunct, that is, there do not exist relations of supply and use between them. A rather natural measure of aggregate value-added based TFP change is therefore ln IV AT F P ≡

K X k=1

L(V Ak1 , V Ak0 ) ln IV AT F P k , k1 , V Ak0 ) L(V A k=1

PK

(22)

where each unit’s TFP change is weighted with its share of (averaged over the two periods) value added in the total value added of the ensemble. Now, if for each unit at both time periods revenue equals cost, that is rkt = ckt (k = 1, ..., K; t = 0, 1), then one obtains immediately by using Theorem 1, ln IV AT F P ≡

K X k=1

L(rk1 , rk0 ) ln IGOT F P ∗k , k1 , V Ak0 ) L(V A k=1

PK

12

(23)

which is a representation of the famous Domar aggregation rule. Each unit’s gross-output based TFP index is weighted by the ratio of its revenue to total value added of the ensemble. Note again that this result was obtained without requiring any economic theory. Put otherwise, this proof of the Domar aggregation rule is much simpler than all the proofs figuring in the literature.8

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Difference type measures

The natural starting point for the gross-output based TFP indicator is the profit difference (r1 − c1 ) − (r0 − c0 ). Due to its linear structure, we have (r1 − c1 ) − (r0 − c0 ) = (r1 − r0 ) − (c1 − c0 ) = (r1 − r0 ) − (c1KL − c0KL ) − (c1M − c0M ).

(24)

It is assumed that the revenue difference can be decomposed into two parts, r1 − r0 = Po (p1 , y 1 , p0 , y 0 ) + Qo (p1 , y 1 , p0 , y 0 ),

(25)

where Po (.) is a price indicator and Qo (.) is a quantity indicator, both operating on the output prices and quantities of both periods. Likewise, it is assumed that the primary input cost difference can be decomposed as 1 0 1 0 c1KL − c0KL = Pi0 (wKL , x1KL , wKL , x0KL ) + Q0i (wKL , x1KL , wKL , x0KL ),

(26)

and that the intermediate input cost difference can be decomposed as 1 0 1 0 c1M − c0M = Pi00 (wM , x1M , wM , x0M ) + Q00i (wM , x1M , wM , x0M ),

(27)

where Pi0 (.) and Pi00 (.) are input price indicators, and Q0i (.) and Q00i (.) are input quantity indicators. The gross-output based indicator of TFP is then defined by 8

See e.g. Aulin-Ahmavaara (2003). This article primarily discusses the problems connected with using the SNA93 accounts as a framework for productivity measurement.

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∆GOT F P ≡ (28) 1 1 0 0 0 1 1 0 0 00 1 1 0 0 Qo (p , y , p , y ) − Qi (wKL , xKL , wKL , xKL ) − Qi (wM , xM , wM , xM ). The natural starting point for the value-added based TFP indicator is the value added minus primary input cost difference (V A1 − c1KL ) − (V A0 − c0KL ). Due to the definition of value added, this can be rewritten as (V A1 − c1KL ) − (V A0 − c0KL ) = (r1 − c1M − c1KL ) − (r0 − c0M − c0KL ) = (r1 − r0 ) − (c1M − c0M ) − (c1KL − c0KL ),

(29)

which is the same decomposition as obtained in (24). Thus the value-added based TFP indicator is identically equal to the gross-output based TFP indicator, ∆V AT F P ≡ ∆GOT F P .

5

The TFP indexes and indicators in continuous time

Instead of considering the time variable t as label of a discrete accounting period, we now consider t as a continuous variable. Accordingly, all prices and quantities are considered as realizations of certain functions of t, so that t ptm = pm (t), ym = ym (t) for m = 1, ..., M , and wnt = wn (t), xtn = xn (t) for n = 1, ..., N . All these functions are assumed to be positive, continuous and piecewise differentiable. Likewise, all bilateral price and quantity indexes will be replaced by Divisia indexes. In particular, the logarithm of the grossoutput based TFP index for period 1 relative to period 0 is defined as Div ln IGOT F P (1, 0) ≡ ln QDiv (1, 0). o (1, 0) − ln Qi

(30)

The Divisia output quantity index is given by ln QDiv o (1, 0)

≡

Z

M 1 X

0 m=1

um (τ )

d ln ym (τ ) dτ, dτ

(31)

where um (τ ) ≡ pm (τ )ym (τ )/p(τ ) · y(τ ) (m = 1, ..., M ) are the output revenue shares. The Divisia input quantity index is given by 14

ln QDiv (1, 0) ≡ i

Z

N 1 X

0 n=1

sn (τ )

d ln xn (τ ) dτ, dτ

(32)

where sn (τ ) ≡ wn (τ )xn (τ )/w(τ )·x(τ ) (n = 1, ..., N ) are the input cost shares. The logarithm of the value-added based TFP index is defined as Div ln IV AT F P (1, 0) ≡ ln QDiv V A (1, 0) − ln QKL (1, 0).

(33)

The Divisia value-added quantity index is given by9 ln QDiv V A (1, 0) ≡ Z 0

1

M X pm (τ )ym (τ ) d ln ym (τ )

v(τ )

m=1

dτ

−

X wn (τ )xn (τ ) d ln xn (τ ) n∈M

v(τ )

dτ

!

dτ,(34)

where v(τ ) ≡ p(τ ) · y(τ ) − wM (τ ) · xM (τ ) is period τ value added. The Divisia primary input quantity index is given by ln QDiv KL (1, 0) ≡

Z

1

X wn (τ )xn (τ ) d ln xn (τ )

0 n∈KL

cKL (τ )

dτ

dτ,

(35)

where cKL (τ ) ≡ wKL (τ ) · xKL (τ ) is period τ primary input cost. In all expressions the symbol τ is employed as generic time variable. Note that the indexes (31), (32), (34), and (35) will in general depend on the path taken by (p(τ ), y(τ ), w(τ ), x(τ )) when τ runs from 0 to 1. Hence, the TFP indexes will in general also be path-dependent. If revenue r(τ ) = p(τ ) · y(τ ) equals cost c(τ ) = w(τ ) · x(τ ), then the Divisia gross-output based TFP index can be written as ln IGOT F P (1, 0) = Z 0

1

1 r(τ )

"

M X m=1

pm (τ )ym (τ )

(36) N X

#

d ln xn (τ ) d ln ym (τ ) − dτ. wn (τ )xn (τ ) dτ dτ n=1

The assumption that revenue equals cost is tantamount to the assumption that value added v(τ ) equals primary input cost cKL (τ ). But then it appears that the Divisia value-added based TFP index can be written as 9

Notice that in this and subsequent formulas the symbol M occurs with two, different, meanings: first, as designating the highest numbered output commodity, and, second, as designating the set of intermediate inputs. The risk of confusion is, however, negligible.

15

ln IV AT F P (1, 0) = Z 0

1

1 v(τ )

"

M X

(37)

pm (τ )ym (τ )

m=1

N X

#

d ln xn (τ ) d ln ym (τ ) wn (τ )xn (τ ) − dτ. dτ dτ n=1

The two expressions (36) and (37) differ by the factor to the left of the bracketed part. Since these factors occur behind the integral sign, they cannot be factored out. By a fundamental property of integrals, however, one obtains ln IV AT F P (1, 0) =

Z

1

0

r(τ ) d ln IGOT F P (τ, 0) dτ. v(τ ) dτ

(38)

If the growth rate d ln IGOT F P (τ, 0)/dτ ≥ 0, then, by the Mean Value Theorem of integral calculus, ln IV AT F P (1, 0) =

r(τ ∗ ) ln IGOT F P (1, 0) v(τ ∗ )

(39)

for some τ ∗ ∈ [0, 1]. The same relation can be obtained under the assumption that the growth rate d ln IV AT F P (τ, 0)/dτ ≥ 0. We see the Domar factor, revenue over value added, again materialize. Expression (39) could be called the continuous time analog of Theorem 1. Turning to indicators, it is immediately clear that N dxn (τ ) dym (τ ) X dτ − wn (τ ) ∆GOT F P (1, 0) = pm (τ ) dτ dτ 0 n=1 m=1 = ∆V AT F P (1, 0), (40)

Z

1

"

M X

#

which mirrors the result obtained in Section 4.

6

Technological change and path-independency

The question is now: what do the two TFP indexes measure? The answer appears to depend on the assumptions one is willing to make. We will employ two structural assumptions and two behavioral assumptions.

16

The first and basic structural assumption concerns the existence of a production technology that smoothly evolves through time. This technology, that governs the unit’s production process at time period τ , is given by S(τ ) ≡ {(x, y) | x can produce y at period τ },

(41)

which is the set of all feasible combinations of input and output quantities. Under weak regularity conditions, the technology can be represented by the cost function C(w, y, τ ) ≡ min {w · x | (x, y) ∈ S(τ )}. x

(42)

This function provides the minimum cost for producing the output quantities y when the input prices are given by w.10 The vector of cost minimizing quantities will be denoted by x(w, y, τ ). Provided that the cost function is continuously differentiable, Shephard’s Lemma tells us that x(w, y, τ ) = ∇w C(w, y, τ ). Technological progress means that the cost of producing y at prices w declines with the mere passage of time. Put otherwise, the cost function based measure of technological change is given by the negative of its first order partial derivative with respect to time, −∂C(w, y, τ )/∂τ , or its growth rate, −∂ ln C(w, y, τ )/∂τ . An other representation of the period τ technology is given by the value added function V A(xKL , wM , p, τ ) ≡ max {p · y − wM · xM | (xKL , xM , y) ∈ S(τ )}. x ,y

(43)

M

This function provides the maximum value added at given output and intermediate input prices, conditional on primary input quantities. This function was introduced and baptized by Diewert (1978b). It exhibits the properties of a restricted profit function, and, provided some prices are allowed to be negative, those of a revenue function. Provided that the value added function is continuously differentiable, Hotelling’s Lemma tells us that the vector of optimal output quantities is given by y(xKL , wM , p, τ ) = ∇p V A(xKL , wM , p, τ ), 10 For the regularity conditions, as well as for alternative representations of the technology, one is referred to F¨ are and Primont (1995).

17

and that the vector of optimal intermediate input quantities is given by xM (xKL , wM , p, τ ) = −∇wM V A(xKL , wM , p, τ ). Technological progress means that, conditional on prices and primary input quantities, value added will increase with the passage of time. Put otherwise, the value added function based measure of technological change is given by its first order partial derivative with respect to time, ∂V A(xKL , wM , p, τ )/ ∂τ , or its growth rate, ∂ ln V A(xKL , wM , p, τ )/∂τ . We will also make use of the so-called indirect cost function IC(wKL , wM /v, p/v, τ ) ≡ min{wKL · xKL | V A(xKL , wM , p, τ ) ≥ v} = xKL

min {wKL · xKL | V A(xKL , wM /v, p/v, τ ) ≥ 1}. x

(44)

KL

This function provides the minimum primary input cost, at given primary input prices, for obtaining maximum value added v, at given intermediate input prices and output prices. Note that the value added function (43) is positively linearly homogeneous in (wM , p), which is why the indirect cost function depends on the variables (wM /v, p/v) rather than (wM , p, v). The indirect cost function is also a representation of the technology S(τ ). Diewert (1978b) called it the unit value added cost function. Shephard’s Lemma tells us that the vector of cost minimizing primary input quantities is given by xKL (wKL , wM /v, p/v, τ ) = ∇wKL IC(wKL , wM /v, p/v, τ ), provided that the indirect cost function is continuously differentiable. In the sequel of this paper we will need expressions for the other partial derivatives of the indirect cost function. The reader may, however, skip the next paragraph for future reference. Application of the Envelope Theorem to expression (44) and usage of the first order conditions for the minimization problem defining the indirect cost function yields11 ∂V A(x∗KL ,wM /v,p/v,τ )

(wn /v) ∂ ln IC(wKL , wM /v, p/v, τ ) ∂(wn /v) (n ∈ M ), =− ∂ ln(wn /v) V A (x∗KL , wM /v, p/v, τ ) 11

See Balk (1998; 132) for the details of the derivation.

18

(45)

∂V A(x∗

,w

/v,p/v,τ )

KL M (pm /v) ∂ ln IC(wKL , wM /v, p/v, τ ) ∂(pm /v) (m = 1, ..., M ) =− ∗ ∂ ln(pm /v) V A (xKL , wM /v, p/v, τ ) (46) ∂V A(x∗KL ,wM /v,p/v,τ ) ∂ ln IC(wKL , wM /v, p/v, τ ) ∂τ , (47) =− ∂τ V A (x∗KL , wM /v, p/v, τ ) where x∗KL is the solution to the minimization problem defining the indirect cost function IC(wKL , wM /v, p/v, τ ), and the value added function based measure of local scale elasticity is defined as

V A (xKL , wM /v, p/v, τ ) ≡

xKL · ∇xKL V A(xKL , wM /v, p/v, τ ) . V A(xKL , wM /v, p/v, τ )

(48)

Since Hotelling’s Lemma implies that ∂V A(x∗KL , wM /v, p/v, τ ) = −xn (x∗KL , wM /v, p/v, τ ) (n ∈ M ) ∂(wn /v) ∂V A(x∗KL , wM /v, p/v, τ ) = ym (x∗KL , wM /v, p/v, τ ) (m = 1, ..., M ), ∂(pm /v) expressions (45) and (46) can be simplified to

(49) (50)

(wn /v)xn (x∗KL , wM /v, p/v, τ ) ∂ ln IC(wKL , wM /v, p/v, τ ) = (n ∈ M ) (51) ∂ ln(wn /v) V A (x∗KL , wM /v, p/v, τ ) ∂ ln IC(wKL , wM /v, p/v, τ ) (pm /v)ym (x∗KL , wM /v, p/v, τ ) =− (m = 1, ..., M ). ∂ ln(pm /v) V A (x∗KL , wM /v, p/v, τ ) (52) We will return to these expressions at a later stage. What is the relation between the two measures of technological change, the cost function based measure and the value added function based measure? Duality theory tells us that the cost function and the value added function are linked to each other via the profit function, in the following way: Π(w, p, τ ) ≡ max{p · y − w · x | (x, y) ∈ S(τ )} x,y

= max{p · y − C(w, y, τ )} y

= max{V A(xKL , wM , p, τ ) − wKL · xKL }. xKL

19

(53)

By the Envelope Theorem one concludes that the two measures of technological change are linked in a simple way, namely by ∂C(w, y ∗ , τ ) ∂Π(w, p, τ ) = − ∂τ ∂τ ∂V A(x∗KL , wM , p, τ ) = , ∂τ

(54)

where y ∗ is the solution to the second maximization problem, and x∗KL is the solution to the third maximization problem of (53). By rearranging expression (54) one obtains for the growth rates that C(w, y ∗ , τ ) ∂ ln C(w, y ∗ , τ ) ∂ ln V A(x∗KL , wM , p, τ ) =− . ∂τ V A(x∗KL , wM , p, τ ) ∂τ

(55)

Note that the factor linking both growth rates resembles the Domar factor.

6.1

The Divisia gross-output based TFP index

We now turn to the interpretation of the Divisia gross-output based TFP index (30).12 The first behavioral assumption is that at each period the actual input quantities are equal to the cost minimizing quantities, given actual input prices and output quantities. That is, the curves x(τ ), w(τ ), and y(τ ) are related by x(τ ) = x(w(τ ), y(τ ), τ ).

(56)

Multiplying both sides of this expression by w(τ ) yields w(τ ) · x(τ ) = C(w(τ ), y(τ ), τ ),

(57)

which means that at any period the actual cost equals the minimum cost for producing the actual output quantities at the actual input prices. This is also expressed by saying that the production unit is cost efficient. Since all functions occurring in expression (57) are continuous, we can differentiate this expression with respect to the time variable τ . The left hand side of (57) yields 12

The following results are well known. The presentation draws on material from Balk (2000).

20

N N d ln(w(τ ) · x(τ )) X d ln wn (τ ) X d ln xn (τ ) = + , sn (τ ) sn (τ ) dτ dτ dτ n=1 n=1

(58)

whereas the right hand side yields d ln C(w(τ ), y(τ ), τ ) = dτ N X ∂ ln C(w(τ ), y(τ ), τ ) d ln wn (τ ) + ∂ ln wn dτ n=1 M X ∂ ln C(w(τ ), y(τ ), τ ) d ln ym (τ )

∂ ln ym ∂ ln C(w(τ ), y(τ ), τ ) . ∂τ

m=1

dτ

+ (59)

The assumption of cost efficiency implies that the actual cost shares of the inputs are equal to the optimal cost shares, which leads to the conclusion that ∂ ln C(w(τ ), y(τ ), τ ) = sn (τ ) (n = 1, ..., N ), (60) ∂ ln wn that is, the partial derivatives occurring in the first term at the right hand side of expression (59) are equal to the actual cost shares of the inputs. The second behavioral assumption is that at any period the output prices are proportional to marginal cost, that is pm (τ ) ∝

∂C(w(τ ), y(τ ), τ ) (m = 1, ..., M ). ∂ym

(61)

Simple manipulations with this relation lead to the following expression: M X ∂ ln C(w(τ ), y(τ ), τ ) ∂ ln C(w(τ ), y(τ ), τ ) = um (τ ) (m = 1, ..., M ), (62) ∂ ln ym ∂ ln ym m=1

where we recall that um (τ ) (m = 1, ..., M ) denote the actual revenue shares of the outputs. 21

The third assumption is again a structural one. It says that the technology exhibits globally constant returns to scale. This implies that M X ∂ ln C(w(τ ), y(τ ), τ ) m=1

∂ ln ym

= 1.

(63)

Hence, the second and third assumption together imply that ∂ ln C(w(τ ), y(τ ), τ ) = um (τ ) (m = 1, ..., M ), ∂ ln ym

(64)

that is, the partial derivatives occurring in the second term at the right hand side of expression (59) are equal to the actual revenue shares of the outputs. We are now in the position to put the various pieces together. By construction the right hand sides of expressions (58) and (59) are identically equal. Thus, substituting the two results (60) and (64), we get the following expression: M X m=1

um (τ )

N ∂ ln C(w(τ ), y(τ ), τ ) d ln xn (τ ) d ln ym (τ ) X − =− . sn (τ ) dτ dτ ∂τ n=1

(65)

After integrating over [0, 1] we obtain, using the definitions (30), (31) and (32) respectively, 1

∂ ln C(w(τ ), y(τ ), τ ) dτ. (66) ∂τ 0 The right hand side of this expression is the integral of the cost function based measure of technological change. If the cost function can be assumed to have the following form, ln IGOT F P (1, 0) = −

Z

˜ C(w, y, τ ) = A(τ )C(w, y),

(67)

that is, technological change is Hicks (input) neutral, then its partial derivative with respect to τ is equal to ∂ ln A(τ ) ∂ ln C(w, y, τ ) = . ∂τ ∂τ In this case expression (66) reduces to

22

(68)

(

1

)

∂ ln A(τ ) IGOT F P (1, 0) = exp − dτ ∂τ 0 = exp{− ln A(1) + ln A(0)} = A(0)/A(1), Z

(69)

that is, the Divisia gross-output based TFP index is path-independent. NoDiv tice that this result does not imply that QDiv (1, 0) are patho (1, 0) and Qi independent. Only their ratio is path-independent. It is useful to recapitulate the assumptions made in deriving (66). First, the production unit is assumed to be cost efficient, that is, technically efficient – it always operates at the frontier of the current technology – as well as allocatively efficient – the input quantities always have the optimal mix. Second, the unit is assumed to act in a competitive environment. Third, the technology is assumed to exhibit globally constant returns to scale. Under these classical assumptions, the Divisia gross-output based TFP index (30) measures technological change. Moreover, if technological change exhibits Hicks (input) neutrality, then the Divisia gross-output based TFP index will be path-independent. It is simple to verify that the first two assumptions follow from the assumption of profit maximization. If p(τ )·y(τ )−w(τ )·x(τ ) = Π(w(τ ), p(τ ), τ ), then the first order condition of the second maximization problem in expression (53) implies that output prices are equal to marginal cost, which is a specific case of (61). Furthermore, it must then be the case that w(τ )·x(τ ) = C(w(τ ), y(τ ), τ ), which in turn implies expression (56). Finally, the assumption of profit maximization combined with the assumption of globally constant returns to scale implies that maximum profit is equal to zero, from which it follows that revenue r(τ ) = p(τ ) · y(τ ) is equal to cost c(τ ) = w(τ ) · x(τ ). When one of these assumptions does not hold, then IGOT F P (1, 0) does not measure pure technological change. For instance, when the technology does not exhibit constant returns to scale, then it is easily seen by reworking the previous reasoning that IGOT F P (1, 0) measures the combined effect of technological change and scale. It is interesting to consider the single output case, i.e. M = 1. The assumption of cost efficiency leads again to expression (60). Instead of expression 23

(64), however, one obtains – without invoking any assumption – ∂ ln C(w(τ ), y(τ ), τ ) M C(τ ) = , ∂ ln y AC(τ )

(70)

where M C(τ ) ≡ ∂C(w(τ ), y(τ ), τ )/∂y denotes marginal cost, and AC(τ ) ≡ C(w(τ ), y(τ ), τ )/y(τ ) denotes average cost. Hence, instead of expression (65), one obtains N d ln xn (τ ) ∂ ln C(w(τ ), y(τ ), τ ) M C(τ ) d ln y(τ ) X sn (τ ) − =− . AC(τ ) dτ dτ ∂τ n=1

(71)

If one additionally assumes that the technology exhibits globally constant returns to scale, then M C(τ ) = AC(τ ), and expression (71) reduces to N d ln xn (τ ) ∂ ln C(w(τ ), y(τ ), τ ) d ln y(τ ) X sn (τ ) − =− . dτ dτ ∂τ n=1

(72)

The left hand side of this expression is frequently used as model for the value-added based TFP index. The single output y(τ ) is then identified as real value added, and the inputs are restricted to be the primary ones. This model, however, presuppposes that the price component of value added is something that is exogenously given. However, this price component depends on quantities of output and intermediate input which are chosen by the unit. In the next section a more realistic approach will be considered.

6.2

The Divisia value-added based TFP index

The question to be answered in this section is: what does the Divisia valueadded based TFP index (33) measure? And, again, the answer depends on the assumptions one is willing to make. Our first assumption now is that actual primary input quantities are optimal under actual primary input prices for producing actual value added. Formally, it is assumed that the curves xKL (τ ), w(τ ), p(τ ), and v(τ ) are related by xKL (τ ) = xKL (wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ). Multiplying both sides of this expression by wKL (τ ) yields 24

(73)

cKL (τ ) = wKL (τ ) · xKL (τ ) = IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ), (74) which means that at any period actual primary input cost equals the minimum cost which is necessary for obtaining actual value added, given actual input and output prices. Since all functions occurring in expression (74) are continuous, we can differentiate this expression with respect to the time variable τ . The left hand side of (74) yields X wn (τ )xn (τ ) d ln wn (τ ) X wn (τ )xn (τ ) d ln xn (τ ) d ln cKL (τ ) = + , dτ cKL (τ ) dτ cKL (τ ) dτ n∈KL n∈KL (75) whereas the right hand side yields

d ln IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) = dτ X ∂ ln IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) d ln wn (τ ) + ∂ ln wn dτ n∈KL X ∂ ln IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) d ln(wn (τ )/v(τ )) n∈M M X

∂ ln(wn /v)

dτ

+

∂ ln IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) d ln(pm (τ )/v(τ )) + ∂ ln(pm /v) dτ m=1 ∂ ln IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) . (76) ∂τ

The first assumption implies that the actual primary input cost shares are equal to their optimal counterparts, which in turn leads to the conclusion that wn (τ )xn (τ ) ∂ ln IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) (n ∈ KL). (77) = ∂ ln wn cKL (τ ) Our second assumption is that, given the actual primary input quantities, the actual output quantities as well as the actual intermediate input quantities are optimal, that is 25

xM (τ ) = xM (xKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ )

(78)

y(τ ) = y(xKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ).

(79)

Our third assumption is that the technology exhibits globally constant returns to scale. It is straightforward to show that this implies that the value added function V A(xKL , wM , p, τ ) is positively linearly homogeneous in xKL .13 This in turn implies that V A (xKL , wM /v, p/v, τ ) is identically equal to 1. Using (51), (52), and (47), the second and third assumption together imply that wn (τ )xn (τ ) ∂ ln IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) = (n ∈ M ) ∂ ln(wn /v) v(τ )

(80)

pm (τ )ym (τ ) ∂ ln IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) =− (m = 1, ..., M ) ∂ ln(pm /v) v(τ ) (81) ∂ ln IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) = ∂τ ∂V A(xKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) − . (82) ∂τ Since by the second assumption v(τ ) = V A(xKL (τ ), wM (τ ), p(τ ), τ ), expression (82) can be written as ∂ ln IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) = ∂τ ∂ ln V A(xKL (τ ), wM (τ ), p(τ ), τ ) − . ∂τ

(83)

13

The technology S(τ ) exhibits globally constant returns to scale if, for all θ > 0, (θx, θy) ∈ S(τ ) whenever (x, y) ∈ S(τ ). Now, by (43), for all θ > 0, V A(θxKL , wM , p, τ )

= =

max{p · y − wM · xM | (θxKL , xM , y) ∈ S(τ )}

xM ,y

max{p · y − wM · xM | (xKL , xM /θ, y/θ) ∈ S(τ )}

xM ,y

= θ

max {p · y/θ − wM · xM /θ | (xKL , xM /θ, y/θ) ∈ S(τ )}

xM /θ,y/θ

= θV A(xKL , wM , p, τ ), which means that the value added function is positively linearly homogeneous in xKL .

26

By substituting (77), (80), (81), and (83) into (76) and equating the result to (75), one obtains X wn (τ )xn (τ ) d ln xn (τ ) n∈KL

cKL (τ )

dτ

=

X wn (τ )xn (τ ) d ln(wn (τ )/v(τ ))

v(τ )

n∈M M X

dτ

pm (τ )ym (τ ) d ln(pm (τ )/v(τ )) v(τ ) dτ m=1 ∂ ln V A(xKL (τ ), wM (τ ), p(τ ), τ ) , − ∂τ −

(84)

or, by rearranging, M X wn (τ )xn (τ ) d ln wn (τ ) X d ln v(τ ) pm (τ )ym (τ ) d ln pm (τ ) − − dτ v(τ ) dτ v(τ ) dτ m=1 n∈M

"

−

X wn (τ )xn (τ ) d ln xn (τ ) n∈KL

cKL (τ )

dτ

=

#

∂ ln V A(xKL (τ ), wM (τ ), p(τ ), τ ) . ∂τ

(85)

Using the definition of value added, this expression can be rewritten as "

M X pm (τ )ym (τ ) d ln ym (τ ) m=1

−

v(τ )

dτ

X wn (τ )xn (τ ) d ln xn (τ ) n∈KL

cKL (τ )

dτ

−

X wn (τ )xn (τ ) d ln xn (τ ) n∈M

=

v(τ )

#

dτ

∂ ln V A(xKL (τ ), wM (τ ), p(τ ), τ ) . ∂τ

(86)

After integrating over [0, 1] and using (33) we obtain 1

∂ ln V A(xKL (τ ), wM (τ ), p(τ ), τ ) dτ, (87) ∂τ 0 which is the integral of the value added function based measure of technological change. If the value added function can be assumed to exhibit the following form, ln IV AT F P (1, 0) =

Z

V A(xKL , wM , p, τ ) = B(τ )V˜A(xKL , wM , p), 27

(88)

that is, technological change is Hicks (value added) neutral, then its partial derivative with respect to τ is equal to ∂ ln V A(xKL , wM , p, τ ) ∂ ln B(τ ) = . ∂τ ∂τ In this case expression (87) reduces to (Z

1

(89)

)

∂ ln B(τ ) IV AT F P (1, 0) = exp dτ ∂τ 0 = exp{ln B(1) − ln B(0)} = B(1)/B(0),

(90)

that is, the Divisia value-added based TFP index is path-independent. NoDiv tice that this result does not imply that QDiv V A (1, 0) and QKL (1, 0) are pathindependent. Only their ratio is path-independent. It is useful to recapitulate the assumptions made in deriving (87). First, the production unit is cost efficient with respect to primary inputs and value added. Put otherwise, the unit ’produces’ value added in a cost efficient way. Second, conditional on its primary input quantities, the unit’s value added is maximal. Third, the technology is assumed to exhibit globally constant returns to scale. Under these assumptions, the Divisia value-added based TFP index (33) measures technological change. Moreover, if technological change exhibits Hicks (value added) neutrality, then this index is path-independent. It is simple to verify that the first two assumptions follow from the assumption of profit maximization. By duality we have also Π(w, p, τ ) = max {v − IC(wKL , wM /v, p/v, τ )}. v

(91)

Hence, if profit p(τ )·y(τ )−wM (τ )·xM (τ )−wKL (τ )·xKL (τ ) = v(τ )−wKL (τ )· xKL (τ ) = Π(wKL (τ ), wM (τ ), p(τ ), τ ), then primary input cost cKL (τ ) = wKL (τ ) · xKL (τ ) = IC(wKL (τ ), wM (τ )/v(τ ), p(τ )/v(τ ), τ ) and value added v(τ ) = V A(xKL (τ ), wM (τ ), p(τ ), τ ). Thus expressions (73), (78), and (79) must also hold. Finally, the assumption of profit maximization combined with the assumption of globally constant returns to scale implies that v(τ ) = cKL (τ ), since maximum profit is equal to zero. 28

6.3

Comparison

We will repeat and compare here the main conclusions of the previous two subsections. If the technology exhibits globally constant returns to scale and the production unit maximizes profit, then the Divisia gross-output based TFP index reduces to a measure of technological change, 1

∂ ln C(w(τ ), y(τ ), τ ) dτ, (92) ∂τ 0 and the value-added based index reduces to a dual measure of technological change, ln IGOT F P (1, 0) = −

Z

1

∂ ln V A(xKL (τ ), wM (τ ), p(τ ), τ ) dτ. ∂τ 0 By using relation (55), one obtains ln IV AT F P (1, 0) =

Z

ln IV AT F P (1, 0) Z 1 C(w(τ ), y(τ ), τ ) ∂ ln C(w(τ ), y(τ ), τ ) = − dτ ∂τ 0 V A(xKL (τ ), wM (τ ), p(τ ), τ ) Z 1 r(τ ) ∂ ln C(w(τ ), y(τ ), τ ) = − dτ ∂τ 0 v(τ ) r(τ ∗ ) = ln IGOT F P (1, 0), v(τ ∗ )

(93)

(94)

where the last step is based on the Mean Value Theorem of integral calculus and τ ∗ ∈ [0, 1]. It is thereby assumed that ∂ ln C(w(τ ), y(τ ), τ )/∂τ ≤ 0. We see the Domar factor again pop up, evaluated at some time period between 0 and 1. Note that, since τ ∗ depends on the path, if IGOT F P (1, 0) is pathindependent, then IV AT F P (1, 0) not, and vice versa. Note further that expression (94) holds for the true, but unknown technology. As soon as one approximates this technology by some specific functional form for the cost or value added function, the exact relationship will get lost. Moreover, the practical necessity of replacing a continuous time path by some discrete approximation tends to introduce additional discrepancies. A paper by Slade (1988) sheds light on this issue.14 14

The first part of Slade’s paper is concerned with finding conditions for the equality

29

In her paper, Slade assumed that the (one output, three inputs) technology is given by a linearly homogeneous translog cost function that exhibits non-neutral technological change. Eight specifications were considered, characterized by different assumptions on the rate of technological change and different assumptions on the substitutability of capital, labour, and materials. Two economic environments were considered, characterized by different trends of materials prices. These assumptions, together with assumed probability distributions for the remaining prices, served to generate the data. The number of time periods considered was 21. For any pair of adjacent periods IV AT F P and IGOT F P were computed according to the T¨ornqvist formula. All in all, Slade considered 16 Monte Carlo experiments. Averaged over replications and time periods it turned out that ln IV AT F P was larger (smaller) than ln IGOT F P in 9 (7) cases. In 4 cases ln IV AT F P turned out to be smaller than 0. A regression of the average values of ln IV AT F P on those of ln IGOT F P led to a coefficient that roughly corresponded to the inverse of the share of value added in total cost, i.e. the Domar factor.

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Conclusion

What is the practical worth of all these theoretical results? I conclude with pointing out some possible applications. 1. If all the data on input and output prices and quantities are available, then the value-added based as well as the gross-output based TFP index can be calculated. Usually, however, only (approximate) results on one of these measures are reported. The theory presented in section 3 provides the means to align results coming from different sources and using different concepts, given that in any case the Domar factor can be computed. 2. Studies using enterprise level data usually report gross-output based TFP index numbers. Trying to compare these to meso or macro figures one runs into the difficulty that those figures are based on the value added output concept. Again, the theory of section 3 makes it possible to turn value-added based TFP index numbers, at least approximately, into gross-output based index numbers, so that these can be used as benchmarks in microdata studies. of IV AT F P (1, 0) and IGOT F P (1, 0). In view of our results, however, this is a fruitless endeavour, and, indeed, the conditions obtained turn out to be quite heroic.

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3. It is well known that outsourcing, which conceptually means that elements of the primary input vector L are transferred to the intermediate input vector M , influences the value-added based TFP index. The grossoutput based TFP index, however, is invariant with respect to outsourcing. If only value-added based TFP index numbers are reported and the extent of outsourcing differs or changes through time, Domar factor values can be used to ’correct’ these index numbers to outsourcing-invariant ones. 4. Even if all the classical assumptions could be assumed to hold, the value-added based and the gross-output based TFP index would provide two, dual but numerically different, measures of technological change. This does not imply a break-down of measurement, but reflects a structural state of affairs. In terms of the model used in this paper, technological change means that the production possibilities set S(τ ), which is a subset of the N + M -dimensional space, moves through time. Measurement means that this movement must be mapped into 1-dimensional space. Clearly, there is no unique way to do this. The best one can hope for are simple rules to transform the various measures into each other. The Domar factor is one of those transformers.

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