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Value data and the Bennet price and quantity indicators
Economics Letters 102 (2009) 19–21
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o n b a s e
Value data and the Bennet price and quantity indicators Robin M. Cross ⁎, Rolf Färe Oregon State University, USA
a r t i c l e
i n f o
Article history: Received 1 October 2007 Received in revised form 3 October 2008 Accepted 7 October 2008 Available online 28 October 2008
a b s t r a c t We derive value-based Bennet indicators, expressed in terms of revenues and expenditures (values) and relative prices, rather than prices and quantities. We then show how relative price information is recoverable from value data when agents are rational. Published by Elsevier B.V.
Keywords: Bennet indicators Prices Weak axiom JEL classificaton: C43 C61 D24 D60
Bennet (1920) showed how a change in social welfare can be decomposed into the sum of a price and quantity change indicator1. During this same period, Fisher (1921) developed his celebrated ratiobased index number approach. In contrast to Bennet, Fisher (1922) provided an axiomatic framework for his indices, now referred to as the test approach. He showed how an index can be derived from, and thus satisfy, a set of desirable numerical properties. Later, Konüs (1939) provided an economic framework for index numbers, showing how certain indices can be derived directly from utility maximization problems. Bennet's indicators, lacking these early foundations, remained relatively obscure. Recent theoretical results, however, have recast Bennet's indicators in promising new light. Chambers (2001)2 provided a new economic framework, using Diewert's (1976) quadratic lemma to derive the indicator from a translation-homothetic utility function. Balk et al. (2004) derived exact relationships between indicators and directional distance functions. Finally, Diewert (2005) developed an additive test approach framework.
Bennet's indicators are calculated using firm-level price and quantity data. Unfortunately, some economic data is available only in terms of revenues and expenditures (values). Value data is convenient, enabling aggregation across dissimilar cost or revenue categories. For-profit firms regularly generate financial performance data in value terms for income tax reporting and lending disclosure. Heavily regulated industries, such as utilities, generate high quality price and quantity data. But, these data too include value elements, for example labor costs. In this article, we extend Bennet price and quantity indicators to this important source of economic data. First, we introduce equivalent value-based Bennet indicators, expressed in terms of firm-level relative prices and values, in place of explicit prices and quantities. We then show how relative price information is directly recoverable from value data when agents are rational. Finally, we provide a simple illustration and conclude. 1. Value-based measures 1.1. Indicators
⁎ Corresponding author. 221B Ballard Hall, Corvallis, OR 97338, USA. Tel.: +1 541 737 1397; fax: +1 541 737 2563. E-mail address: [email protected] (R.M. Cross). 1 We follow Diewert (2005) and call index numbers that are based on differences indicators. 2 Chambers (2001) termed it as the Bennet-Bowley measure, referring to Bennet (1920) and Bowley (1928).
0165-1765/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.econlet.2008.10.003
Following the notation of Diewert (2005), when prices and quantities (net puts) fpt ; yt gaRM+ + are known for M goods and time periods t = 0,1, the Bennet decomposition is p1 y1 −p0 y0
=
I
+
V;
ð1Þ
20
R.M. Cross, R. Färe / Economics Letters 102 (2009) 19–21
where the price indicator is I
=
1 1 y + y0 p1 −p0 ; 2
ð2Þ
and the quantity indicator is V
=
1 1 p + p0 y1 −y0 : 2
ð3Þ
Suppose only value data fvt gaRM are available, where v tm = P tmY tm, for goods m = 1,…,M. We wish to derive an expression equivalent to the Bennet price indicator, but expressed in terms of M values. Distributing terms, Eq. (2) can be re-expressed in summation notation I
=
1 M 1 1 0 1 ∑ p y −p y + p1m y0m −p0m y0m : 2m=1 m m m m
ð4Þ
0 1 Let relative price z 01 m = p m/p m, which we assume constant across firms. Substituting, we obtain the value-based price indicator
I
=
1 M 0 10 1 01 ∑ v z −1 −vm zm −1 : 2m=1 m m
ð5Þ
The value-based quantity indicator is derived mutatis mutandis. Fig. 2. Price ration constraints rationalize value data.
1.2. Axioms It may happen that neither explicit nor relative price information is available. In that case, relative price information is recoverable from value data when agents are rational. By rationality, we mean that agents satisfy the Weak Axiom of Profit Maximization, given by Varian (1984)
The value-based Weak Axiom of Profit Maximization, written in terms of values and relative price ratios, follows directly from the Eq. (6), iv1 zz10 v0
and
iv0 z z01 v1 :
ð8Þ
M
p1 y1 zp1 y0
and p0 y0 zp0 y1 :
ð6Þ
Price constraints imposed by the Weak Axiom, in terms of displacement vectors Δy = y1 − y0, are p1 Δy z
0
z −p0 Δy:
ð7Þ
Fig. 1 illustrates a convex price cone that rationalizes the data in a simple two period, two good, three firm model. Quantity changes for firm k are labeled Δyk. Firms that lie within or along the cone (doubleshaded region) satisfy the Weak Axiom.
Here, i = ½1; 1; N ; 1pR is a conforming vector of ones. The non-linear constraints in the value-based Weak Axiom result from the reciprocal relationship between elements of z10 and z01 and are illustrated in Fig. 2. Price ratios that lie within the double-shaded area A rationalize the value data. The y-axis and x-axis, labeled z1 and z2 respectively, are the first and second good's price ratios. Constraint line Znk corresponds to firm k's nth constraint. 2. Programming problem When the true price ratios are not known, the constraints of the value-based Weak Axiom in Eq. (8) suggest a programming problem to recover relative (shadow) price information. We specify the problem below, for the 2-period, 2-good case with K firms. Let λ be a 2K-vector of distance functions. Because the true price ratios are unavailable, treat z01 as an M-vector of intensity variables. Again, let i denote a conforming ones vector. The estimated relative price vector ˆz 01 then solves the minimization problem: zˆ
01
i vtk −zts vsk zλtk ; k = 1; N ; K; st −1 ; m = 1; N ; M; zts m = zm
= argminf i λ : zts
ð9Þ
t; s = 0; 1;
zts N0; λz0 g The first constraint line in Eq. (9) follows directly from the valuebased Weak Axiom. The second and third constraint lines result from Table 1 Simplified price and quantity data Period 0
Fig. 1. Finite price cone rationalizes (envelopes) quantity data.
Period 1
Quantities
Prices
Firms
1
2
1
2
1
Quantities 2
1
Prices 2
1 2 3
1 2 1
−1 −2 −1
1 1 1
2 2 2
2.9 3 2
−2 −3 −2.9
2 2 2
1 1 1
R.M. Cross, R. Färe / Economics Letters 102 (2009) 19–21
the reciprocal nature of the relative price ratios. When rank conditions are satisfied and a solution exists, we refer to ˆz 01 as the shadow price ratio and illustrate it in a simple example below.
21
greater than the true indicator values. In this way, Bennet's indicators can be computed using only value data. 4. Conclusion
3. Empirical example Consider first a simplified price and quantity-based dataset for three firms, two goods, and two periods: The resulting Bennet price and quantity indicators are 11.9 and 0, 01 respectively. Note the price ratios z01 are 0.5 and 2, 1 and z 2 respectively. All three firms in our example satisfy the Weak Axiom and relative prices are constant across firms. The quantities detailed in Table 1, if graphed as in Fig. 1, would span most of the convex cone rationalized by the true prices. For this reason, we expect the shadow price ratios calculated by the model in Eq. (9) to approximate the true price ratios. If only value data for these firms are available, the researcher observes the relatively sparse table of revenues and expenditures (Table 2). Solving the programming problem for this data, we obtain shadow price ratios of 0.45 and 1.82, for ˆz 01 z 01 1 and ˆ 2 , respectively, both somewhat less than the true price ratios. Using these shadow price ratios, we compute the value-based Bennet price and quantity indicators, as in Eq. (5), which are 11.74 and 0.16, respectively, slightly
Table 2 Corresponding revenue and expenditure data Period 0
Period 1
Firms
Revenues
Expenditures
Revenues
Expenditures
1 2 3
1 2 1
−2 −4 −2
5.8 6 4
−2 −3 − 2.9
Economic data are frequently reported value terms, rather than price and quantity terms. We exploited the Weak Axiom of Profit Maximization to show how to recover relative price information from value data when agents are rational. Adding to the growing body of theoretical work, we extended the Bennet indicators of price and quantity change to value data by introducing the equivalent valuebased indicators and demonstrating their use. We did not explore the econometric properties of the relative price information obtained from the Weak Axiom's non-linear constraints. Nor did we consider value-based applications to more traditional or more computationally expensive social welfare measures. We leave both of these important questions to future research. References Balk, B.M., Färe, R., Grosskopf, S., 2004. The theory of economic price and quantity indicators. Econometric Theory 23, 149–167. Bennet, T.L., 1920. The theory of measurement of changes in cost of living. Journal of the Royal Statistical Society 83 (3), 455–462. Bowley, A.L., 1928. Notes on Index Numbers. The Economic Journal 38, 216–237. Chambers, R.G., 2001. Consumers' surplus as an exact and superlative cardinal welfare indicator. International Economic Review 42 (1), 105–119. Diewert, W.E., 1976. Exact and superlative index numbers. Journal of Econometrics 4, 115–145. Diewert, W.E., 2005. Index number theory using differences rather than ratios. The American Journal of Economics and Sociology 64, 311–360. Fisher, I., 1921. The best form of index number. Journal of the American Statistical Association 17, 533–537. Fisher, I., 1922. The Making of Index Numbers. Houghton-Mifflin, Boston. Konüs, A.A., 1939. The problem of the true index of the cost of living. Econometrica 7, 10–29. Varian, H.R., 1984. The nonparametric approach to production analysis. Econometrica 52 (3), 579–597.
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o n b a s e
Value data and the Bennet price and quantity indicators Robin M. Cross ⁎, Rolf Färe Oregon State University, USA
a r t i c l e
i n f o
Article history: Received 1 October 2007 Received in revised form 3 October 2008 Accepted 7 October 2008 Available online 28 October 2008
a b s t r a c t We derive value-based Bennet indicators, expressed in terms of revenues and expenditures (values) and relative prices, rather than prices and quantities. We then show how relative price information is recoverable from value data when agents are rational. Published by Elsevier B.V.
Keywords: Bennet indicators Prices Weak axiom JEL classificaton: C43 C61 D24 D60
Bennet (1920) showed how a change in social welfare can be decomposed into the sum of a price and quantity change indicator1. During this same period, Fisher (1921) developed his celebrated ratiobased index number approach. In contrast to Bennet, Fisher (1922) provided an axiomatic framework for his indices, now referred to as the test approach. He showed how an index can be derived from, and thus satisfy, a set of desirable numerical properties. Later, Konüs (1939) provided an economic framework for index numbers, showing how certain indices can be derived directly from utility maximization problems. Bennet's indicators, lacking these early foundations, remained relatively obscure. Recent theoretical results, however, have recast Bennet's indicators in promising new light. Chambers (2001)2 provided a new economic framework, using Diewert's (1976) quadratic lemma to derive the indicator from a translation-homothetic utility function. Balk et al. (2004) derived exact relationships between indicators and directional distance functions. Finally, Diewert (2005) developed an additive test approach framework.
Bennet's indicators are calculated using firm-level price and quantity data. Unfortunately, some economic data is available only in terms of revenues and expenditures (values). Value data is convenient, enabling aggregation across dissimilar cost or revenue categories. For-profit firms regularly generate financial performance data in value terms for income tax reporting and lending disclosure. Heavily regulated industries, such as utilities, generate high quality price and quantity data. But, these data too include value elements, for example labor costs. In this article, we extend Bennet price and quantity indicators to this important source of economic data. First, we introduce equivalent value-based Bennet indicators, expressed in terms of firm-level relative prices and values, in place of explicit prices and quantities. We then show how relative price information is directly recoverable from value data when agents are rational. Finally, we provide a simple illustration and conclude. 1. Value-based measures 1.1. Indicators
⁎ Corresponding author. 221B Ballard Hall, Corvallis, OR 97338, USA. Tel.: +1 541 737 1397; fax: +1 541 737 2563. E-mail address: [email protected] (R.M. Cross). 1 We follow Diewert (2005) and call index numbers that are based on differences indicators. 2 Chambers (2001) termed it as the Bennet-Bowley measure, referring to Bennet (1920) and Bowley (1928).
0165-1765/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.econlet.2008.10.003
Following the notation of Diewert (2005), when prices and quantities (net puts) fpt ; yt gaRM+ + are known for M goods and time periods t = 0,1, the Bennet decomposition is p1 y1 −p0 y0
=
I
+
V;
ð1Þ
20
R.M. Cross, R. Färe / Economics Letters 102 (2009) 19–21
where the price indicator is I
=
1 1 y + y0 p1 −p0 ; 2
ð2Þ
and the quantity indicator is V
=
1 1 p + p0 y1 −y0 : 2
ð3Þ
Suppose only value data fvt gaRM are available, where v tm = P tmY tm, for goods m = 1,…,M. We wish to derive an expression equivalent to the Bennet price indicator, but expressed in terms of M values. Distributing terms, Eq. (2) can be re-expressed in summation notation I
=
1 M 1 1 0 1 ∑ p y −p y + p1m y0m −p0m y0m : 2m=1 m m m m
ð4Þ
0 1 Let relative price z 01 m = p m/p m, which we assume constant across firms. Substituting, we obtain the value-based price indicator
I
=
1 M 0 10 1 01 ∑ v z −1 −vm zm −1 : 2m=1 m m
ð5Þ
The value-based quantity indicator is derived mutatis mutandis. Fig. 2. Price ration constraints rationalize value data.
1.2. Axioms It may happen that neither explicit nor relative price information is available. In that case, relative price information is recoverable from value data when agents are rational. By rationality, we mean that agents satisfy the Weak Axiom of Profit Maximization, given by Varian (1984)
The value-based Weak Axiom of Profit Maximization, written in terms of values and relative price ratios, follows directly from the Eq. (6), iv1 zz10 v0
and
iv0 z z01 v1 :
ð8Þ
M
p1 y1 zp1 y0
and p0 y0 zp0 y1 :
ð6Þ
Price constraints imposed by the Weak Axiom, in terms of displacement vectors Δy = y1 − y0, are p1 Δy z
0
z −p0 Δy:
ð7Þ
Fig. 1 illustrates a convex price cone that rationalizes the data in a simple two period, two good, three firm model. Quantity changes for firm k are labeled Δyk. Firms that lie within or along the cone (doubleshaded region) satisfy the Weak Axiom.
Here, i = ½1; 1; N ; 1pR is a conforming vector of ones. The non-linear constraints in the value-based Weak Axiom result from the reciprocal relationship between elements of z10 and z01 and are illustrated in Fig. 2. Price ratios that lie within the double-shaded area A rationalize the value data. The y-axis and x-axis, labeled z1 and z2 respectively, are the first and second good's price ratios. Constraint line Znk corresponds to firm k's nth constraint. 2. Programming problem When the true price ratios are not known, the constraints of the value-based Weak Axiom in Eq. (8) suggest a programming problem to recover relative (shadow) price information. We specify the problem below, for the 2-period, 2-good case with K firms. Let λ be a 2K-vector of distance functions. Because the true price ratios are unavailable, treat z01 as an M-vector of intensity variables. Again, let i denote a conforming ones vector. The estimated relative price vector ˆz 01 then solves the minimization problem: zˆ
01
i vtk −zts vsk zλtk ; k = 1; N ; K; st −1 ; m = 1; N ; M; zts m = zm
= argminf i λ : zts
ð9Þ
t; s = 0; 1;
zts N0; λz0 g The first constraint line in Eq. (9) follows directly from the valuebased Weak Axiom. The second and third constraint lines result from Table 1 Simplified price and quantity data Period 0
Fig. 1. Finite price cone rationalizes (envelopes) quantity data.
Period 1
Quantities
Prices
Firms
1
2
1
2
1
Quantities 2
1
Prices 2
1 2 3
1 2 1
−1 −2 −1
1 1 1
2 2 2
2.9 3 2
−2 −3 −2.9
2 2 2
1 1 1
R.M. Cross, R. Färe / Economics Letters 102 (2009) 19–21
the reciprocal nature of the relative price ratios. When rank conditions are satisfied and a solution exists, we refer to ˆz 01 as the shadow price ratio and illustrate it in a simple example below.
21
greater than the true indicator values. In this way, Bennet's indicators can be computed using only value data. 4. Conclusion
3. Empirical example Consider first a simplified price and quantity-based dataset for three firms, two goods, and two periods: The resulting Bennet price and quantity indicators are 11.9 and 0, 01 respectively. Note the price ratios z01 are 0.5 and 2, 1 and z 2 respectively. All three firms in our example satisfy the Weak Axiom and relative prices are constant across firms. The quantities detailed in Table 1, if graphed as in Fig. 1, would span most of the convex cone rationalized by the true prices. For this reason, we expect the shadow price ratios calculated by the model in Eq. (9) to approximate the true price ratios. If only value data for these firms are available, the researcher observes the relatively sparse table of revenues and expenditures (Table 2). Solving the programming problem for this data, we obtain shadow price ratios of 0.45 and 1.82, for ˆz 01 z 01 1 and ˆ 2 , respectively, both somewhat less than the true price ratios. Using these shadow price ratios, we compute the value-based Bennet price and quantity indicators, as in Eq. (5), which are 11.74 and 0.16, respectively, slightly
Table 2 Corresponding revenue and expenditure data Period 0
Period 1
Firms
Revenues
Expenditures
Revenues
Expenditures
1 2 3
1 2 1
−2 −4 −2
5.8 6 4
−2 −3 − 2.9
Economic data are frequently reported value terms, rather than price and quantity terms. We exploited the Weak Axiom of Profit Maximization to show how to recover relative price information from value data when agents are rational. Adding to the growing body of theoretical work, we extended the Bennet indicators of price and quantity change to value data by introducing the equivalent valuebased indicators and demonstrating their use. We did not explore the econometric properties of the relative price information obtained from the Weak Axiom's non-linear constraints. Nor did we consider value-based applications to more traditional or more computationally expensive social welfare measures. We leave both of these important questions to future research. References Balk, B.M., Färe, R., Grosskopf, S., 2004. The theory of economic price and quantity indicators. Econometric Theory 23, 149–167. Bennet, T.L., 1920. The theory of measurement of changes in cost of living. Journal of the Royal Statistical Society 83 (3), 455–462. Bowley, A.L., 1928. Notes on Index Numbers. The Economic Journal 38, 216–237. Chambers, R.G., 2001. Consumers' surplus as an exact and superlative cardinal welfare indicator. International Economic Review 42 (1), 105–119. Diewert, W.E., 1976. Exact and superlative index numbers. Journal of Econometrics 4, 115–145. Diewert, W.E., 2005. Index number theory using differences rather than ratios. The American Journal of Economics and Sociology 64, 311–360. Fisher, I., 1921. The best form of index number. Journal of the American Statistical Association 17, 533–537. Fisher, I., 1922. The Making of Index Numbers. Houghton-Mifflin, Boston. Konüs, A.A., 1939. The problem of the true index of the cost of living. Econometrica 7, 10–29. Varian, H.R., 1984. The nonparametric approach to production analysis. Econometrica 52 (3), 579–597.