This document discusses the economic-theoretic approach to index numbers, including output and input price indices. It describes how output price indices can be defined using revenue functions and how they satisfy desirable properties if technology is output-homothetic or exhibits implicit output neutrality. Input price indices are defined using cost functions and satisfy useful properties if technology is input-homothetic or exhibits implicit Hicks neutrality. The Törnqvist and Fisher indices are shown to approximate the theoretical indices under certain conditions.

1. ECON 377/477

2. Topic 5.2Index Numbers (continued)

3. OutlineEconomic-theoretic approachSimple numerical example3ECON377/477 Topic 5.2

4. Economic-theoretic approachThe economic-theoretic approach to index numbers postulates a functional relationship between observed prices and quantities for inputs and outputsIn the case of productivity measurement, the microeconomic theory of the firm is especially relevantWe consider the general case involving M outputs and N inputsECON377/477 Topic 5.24

5. Economic-theoretic approachLet:s and t represent two time periods or firms

6. pms and pmt represent output prices for the m-th commodity in periods s and t, respectively

7. qms and qmt represent output quantities in periods s and t, respectively (m = 1,2,...,M)

8. wns and wnt represent input prices in periods s and t, respectively

9. xns and xnt represent input quantities in periods s and t, respectively (n = 1,2,...,N)ECON377/477 Topic 5.25

10. Economic-theoretic approachFurther, let pt, ps, qt, qs, ws, wt, xt and xs represent vectors of non-negative real numbers of appropriate dimensionsLet Ss and St represent the production technologies in periods s and t, respectivelyIn deriving various price and quantity index numbers for inputs and outputs, we make use of revenue and cost functions, and input and output distance functionsECON377/477 Topic 5.26

11. Economic-theoretic approachThe economic-theoretic approach to index numbers assumes that the firms observed in periods s and t are both technically and allocatively efficientThis means that observed output and input data are assumed to represent optimising behaviour involving revenue maximisation and cost minimisation, or, in some cases, constrained optimisation involving revenue maximisation with cost constraintsECON377/477 Topic 5.27

12. Economic-theoretic approach: output price indicesFor a given level of inputs, x, let the (maximum) revenue function be defined, for technology in period-t, as {pq: (x,q) is feasible in St}The point of tangency between the production possibility curve and the isorevenue line indicates the combination of the two outputs (q1 and q2) that maximise revenue, given the input vector, x, the output price vector, pt, and the technology, StECON377/477 Topic 5.28

13. Economic-theoretic approach: output price indicesy2revenue maximisation given Ptslope =-(p2/p1)St(y,x)0y1ECON377/477 Topic 5.29

14. Economic-theoretic approach: output price indicesThe output price index based on period-t technology, is defined as:This index is the ratio of the maximum revenues possible with the two price vectors, ps and pt, using a fixed level of inputs, x, and period-t technologyThe revenue-maximising points for the price vectors, ptand ps are shown on the next slideECON377/477 Topic 5.210

15. Economic-theoretic approach: output price indicesy2slope =-(pt2/pt1)revenue maximisation given psrevenue maximisation given ptslope =-(ps2/ps1)St(y,x)0y1ECON377/477 Topic 5.211

16. Economic-theoretic approach: output price indicesThe output price index can also be defined using period-s technology leading toThese two price indices depend on whether it is the period-t or period-s technology, and then on the input vector, x, at which the index is calculatedUnder what conditions are the indices independent of these two factors? ECON377/477 Topic 5.212

17. Economic-theoretic approach: output price indicesThese indices are independent of x if and only if the technology is output-homotheticA production technology is output-homothetic if the output sets P(x) depend upon the output set for the unit input vector (input quantities equal to one for all inputs) and a real-valued function, G(x), of xIn simple terms, the production possibility curves for different input vectors, x, are all parallel shifts of the production possibility curve for the unit-input vectorECON377/477 Topic 5.213

18. Economic-theoretic approach: output price indicesIn a similar vein, it can be shown that if the technology exhibits implicit output neutrality, the indices are independent of which period’s technology is used in the derivationThe output price index numbers satisfy the properties of monotonicity, linear homogeneity, identity, proportionality, independence of units of measurement, transitivity for fixed t and x, and time-reversal propertiesECON377/477 Topic 5.214

19. Economic-theoretic approach: output price indicesSince xt and xs are the actual input levels used in periods t and s, we can define the indices using the actual input levels, leading to two natural output price index numbers:ECON377/477 Topic 5.215

20. Economic-theoretic approach: output price indicesWe can get close to the above theoretically defined index numbers in equations in a number of waysUnder the assumptions of allocative and technical efficiency, and regularity conditions on the production technologies, the two index numbers are, respectively, bounded from above and below by the Laspeyres and Paasche indicesA reasonable approximation to the geometric mean of the two indices is provided by the Fisher output price index numberECON377/477 Topic 5.216

21. Economic-theoretic approach: output price indicesAn assumption that the revenue functions have the translog form is in line with the fact that the translog function is a flexible form and provides a second-order approximation to the unknown revenue functionThe translog revenue function is given byECON377/477 Topic 5.217

22. Economic-theoretic approach: output price indicesWe can represent the revenue functions for periods s and t by translog functions, with second-order coefficients being equal for periods s and t (kjt = kjs, mjt = mjs,, kmt = kms)The geometric mean of the two natural output price index numbers is equal to the Törnqvist output price indexECON377/477 Topic 5.218

23. Economic-theoretic approach: output price indicesThe importance of this result is that, even though the theoretical indices require knowledge of the parameters of the revenue function, their geometric mean is equal to the Törnqvist index and the index can be computed from the observed price and quantity dataKnowledge of the parameters of the translog functions is therefore unnecessaryECON377/477 Topic 5.219

24. Economic-theoretic approach: output price indicesThe Törnqvist index is considered to be exact for the translog revenue functionAlso, it is considered superlative since the translog function is a flexible functional formThat is, it provides a second-order approximation to any arbitrary functionThe Fisher index is exact for a quadratic function and, hence, is also superlativeECON377/477 Topic 5.220

25. Economic-theoretic approach: input price indicesWe can measure input price index numbers by comparing costs of producing a vector of outputs, given different input price vectorsWe need to define a cost function, associated with a given production technology, for a given output level, q, namely:The cost function, Ct(w,q), is the minimum cost of producing q, given period-t technology, using the input price vector, wECON377/477 Topic 5.221

26. Economic-theoretic approach: input price indicesWe can use the cost function to define input price index numbersGiven the input prices, wt and ws in periods t and s, we can define the input price index as the ratio of the minimum costs of producing a given output vector q using an arbitrarily selected production technology, Sj (j = s,t)The index is given byECON377/477 Topic 5.222

27. Economic-theoretic approach: input price indicesThe cost elements in the equation on the previous slide can be seen from the diagram on the next slideThe isoquant under technology, Ss, for a given output level, q, is represented by Isoq(q)-SsThe sets of input prices, wsand wt, are represented by isocost lines AA and BBMinimum-cost combinations of inputs producing output vector, q, for these two input price vectors are given by the points, x* and x**ECON377/477 Topic 5.223

28. Economic-theoretic approach: input price indicesx2baBx**Ax*Isoq(q)-Ss0abABx1ECON377/477 Topic 5.224

29. Economic-theoretic approach: input price indicesThese points are obtained by shifting lines AA and BB to aa and bb, respectively, where they are tangential to Isoq(y)-SsThe input price index number for this two input case is then given by the ratio of the costs at points, x* and x**It satisfies many useful properties, including monotonicity, linear homogeneity in input prices independence of units of measurement, proportionality and transitivity (for a fixed q and technology)ECON377/477 Topic 5.225

30. Economic-theoretic approach: input price indicesTo compute the input price index, we need to specify the technology and also the output level, q, at which we wish to compute the indexFirst, the price index is independent of which period technology we use if and only if the technology exhibits implicit Hicks input neutralitySecond, the index, Pi(wt, ws, q), for a given technology is independent of the output level, q, if and only if the technology exhibits input homotheticityMRS between any inputs is independent of the technologyECON377/477 Topic 5.226

31. Economic-theoretic approach: input price indicesIf the technology does not satisfy these conditions, we can define many input price index numbers using alternative specifications for technology, S, and the output vector, qTwo natural specifications are to use the period-s and period-t technologies, along with the output vectors, qs and qtECON377/477 Topic 5.227

32. Economic-theoretic approach: input price indicesThey result in the following input price index numbers:Assuming allocative and technical efficiency, the observed input costs, wsxs and wtxt, are equal to Cs(ws,qs) and Ct(wt,qt), respectively ECON377/477 Topic 5.228

33. Economic-theoretic approach: input price indicesThe Laspeyres and Paasche indices provide upper and lower bounds to the economic-theoretic index numbers in the equations on the previous slideThe geometric mean of these indices can be approximated by the Fisher price index numbers for input pricesAssume the technologies in periods t and s are represented by the translog cost function, with the additional assumption that the second-order coefficients are identical in these periodsECON377/477 Topic 5.229

34. Economic-theoretic approach: input price indicesUnder the assumption of technical and allocative efficiency, the geometric mean of the two input price index numbers in the above equations is given by the Törnqvist price index number applied to input prices and quantitiesThat is, where snt and sns are the input expenditure shares of n-th input in periods t and s, respectively Törnquist indexECON377/477 Topic 5.230

35. Economic-theoretic approach: input price indicesThese results imply that the Fisher and Törnqvist indices can be applied to measure changes in input prices and at the same time have a proper economic-theoretic framework to support their useThey also illustrate that, under certain assumptions, it is not necessary to know the numerical values of the parameters of the cost or revenue function or the underlying production technology; it is sufficient to have the observed input price and quantity data to measure changes in input prices ECON377/477 Topic 5.231

36. Economic-theoretic approach: input price indicesThe Törnqvist input price index is exact for the geometric mean of the two theoretical indices when the underlying cost function is translog, and hence can also be considered superlativeThe Fisher input price index is exact for a quadratic cost functionECON377/477 Topic 5.232

37. Economic-theoretic approach: output quantity indicesUnlike the case of price index numbers, three strategies can be followed in deriving theoretically sound quantity index numbersWe focus on only one, the Malmquist index, which is defined using the distance functionThe Malmquist output index, based on technology in period-t, is defined as: for an arbitrarily selected input vector, xECON377/477 Topic 5.233

38. Economic-theoretic approach: output quantity indicesA similar Malmquist index can be defined using period-s technologyThe index defined on the previous slide is independent of the technology involved if and only if the technology exhibits Hicks output neutralityThe quantity index is independent of the input level, x, if and only if the technology is output homotheticMRT between any inputs is independent of the technologyECON377/477 Topic 5.234

39. Economic-theoretic approach: output quantity indicesEven in the cases where these assumptions hold, we still need to know the functional form of the distance function as well as numerical values of all the parameters involvedThe index number approach attempts to bypass this problem by providing approximations to the index when we are unsure of the functional form, or do not have adequate information to estimate its parameters even when we know the functional formECON377/477 Topic 5.235

40. Economic-theoretic approach: output quantity indicesConsider output quantity indices based on technology in periods s and t, along with the inputs used in these periodsWe have two possible measures of output change, given by Qos(qt, qs, xs) and Qot(qt, qs, xt)There are many standard results that relate these indices to the standard Laspeyres and Paasche quantity index numbersA result of particular interest is that the Fisher index provides an approximation to the geometric average of these two indicesECON377/477 Topic 5.236

41. Economic-theoretic approach: output quantity indicesThe following result establishes the economic-theoretic properties of the Törnqvist output index and shows why the index is considered to be an exact and a superlative indexIf the distance functions for periods s and t are both represented by translog functions with identical second-order parameters, a geometric average of the Malmquist output indices, based on technologies of periods s and t, with corresponding input vectors xs and xt, is equivalent to the Törnqvist output quantity index ECON377/477 Topic 5.237

42. Economic-theoretic approach: output quantity indicesThat is, the Törnqvist output index equals:This result implies that the Törnqvist index is exact for the geometric mean of the period-t and period-s theoretical output index numbers when the technology is represented by a translog output distance functionSince the translog functional form is flexible, the Törnqvist index is also considered to be superlative ECON377/477 Topic 5.238

43. Economic-theoretic approach: input quantity indicesWe describe the input quantity index number, derived using the Malmquist distance measureUsing the concept of the input distance function, we can now define the input quantity index along the same lines as the output indexWe can compare the levels of input vectors xt and xs, by measuring their respective distances from a given output vector for a given state of the production technologyECON377/477 Topic 5.239

44. Economic-theoretic approach: input quantity indicesThe input quantity index based on the Malmquist input distance function is defined for input vectors, xs and xt, with base period-s and using period-t technology:It satisfies monotonocity and linear homogeneity in the input vector xt, is invariant to scalar multiplication of the input vectors, and is independent of units of measurement ECON377/477 Topic 5.240

45. Economic-theoretic approach: input quantity indicesFollowing the same approach as in the previous sections, we note that the input quantity index depends upon the output level, q, we choose, as well as the production technologyIf we use period-s technology in defining the input distance functions, we get the following index:This index and the index on the previous slide are independent of the reference output vector, q, if and only if the technology exhibits input homotheticityECON377/477 Topic 5.241

46. Economic-theoretic approach: input quantity indicesOur main purpose is to relate this Malmquist input quantity index number to the input index number derived using some of the formulae aboveThe input index defined using base period-s technology is bounded from above by the Laspeyres quantity indexThe index defined on current period-t technology, is bounded from below by the Paasche quantity indexECON377/477 Topic 5.242

47. Economic-theoretic approach: input quantity indicesTherefore, the Fisher input quantity index provides an approximation to the geometric mean of the indices, Qis(xt,xs,q) and Qit(xt,xs,q)If we assume a quadratic function, then the Fisher input quantity index can be shown to be equal to the geometric average of the two indicesECON377/477 Topic 5.243

48. Economic-theoretic approach: input quantity indicesIf the distance functions are of the translog form, the distance functions in periods t and s have identical second-order coefficients satisfying the usual restrictions on the parameters of the translog form, and the assumptions of allocative and technical efficiency holds, then the Törnquist input quantity index equalsECON377/477 Topic 5.244

49. Economic-theoretic approach: input quantity indicesIn this equation, snt and sns are input cost-shares in periods t and s, respectivelyThis result shows that the Törnqvist index is exact and superlative for the geometric mean of Malmquist input index numbers based on the technologies of periods t and sRefer to CROB for a simple numerical example involving index numbers on pages 113-115ECON377/477 Topic 5.245

50. Economic-theoretic approach: input quantity indicesIt is easy to see how the index number literature is closely connected with productivity measurementFor example, the Hicks-Moorsteen TFP index, defined as a ratio of the output and input quantity index numbers, can be made operational using the results in this sectionSimilarly, if we wish to use profitability ratios and adjust them for price level differences, we need to make use of appropriate output and input price index numbersECON377/477 Topic 5.246