Topic 5.2
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    Topic 5.2 Topic 5.2 Presentation Transcript

    • ECON 377/477
    • Topic 5.2
      Index Numbers (continued)
    • Outline
      Economic-theoretic approach
      Simple numerical example
      3
      ECON377/477 Topic 5.2
    • Economic-theoretic approach
      The economic-theoretic approach to index numbers postulates a functional relationship between observed prices and quantities for inputs and outputs
      In the case of productivity measurement, the microeconomic theory of the firm is especially relevant
      We consider the general case involving M outputs and N inputs
      ECON377/477 Topic 5.2
      4
    • Economic-theoretic approach
      Let:
      • s and t represent two time periods or firms
      • pms and pmt represent output prices for the m-th commodity in periods s and t, respectively
      • qms and qmt represent output quantities in periods s and t, respectively (m = 1,2,...,M)
      • wns and wnt represent input prices in periods s and t, respectively
      • xns and xnt represent input quantities in periods s and t, respectively (n = 1,2,...,N)
      ECON377/477 Topic 5.2
      5
    • Economic-theoretic approach
      Further, let pt, ps, qt, qs, ws, wt, xt and xs represent vectors of non-negative real numbers of appropriate dimensions
      Let Ss and St represent the production technologies in periods s and t, respectively
      In deriving various price and quantity index numbers for inputs and outputs, we make use of revenue and cost functions, and input and output distance functions
      ECON377/477 Topic 5.2
      6
    • Economic-theoretic approach
      The economic-theoretic approach to index numbers assumes that the firms observed in periods s and t are both technically and allocatively efficient
      This 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 constraints
      ECON377/477 Topic 5.2
      7
    • Economic-theoretic approach: output price indices
      For 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, St
      ECON377/477 Topic 5.2
      8
    • Economic-theoretic approach: output price indices
      y2
      revenue maximisation given Pt
      slope =-(p2/p1)
      St(y,x)
      0
      y1
      ECON377/477 Topic 5.2
      9
    • Economic-theoretic approach: output price indices
      The 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 technology
      The revenue-maximising points for the price vectors, ptand ps are shown on the next slide
      ECON377/477 Topic 5.2
      10
    • Economic-theoretic approach: output price indices
      y2
      slope =-(pt2/pt1)
      revenue maximisation given ps
      revenue maximisation given pt
      slope =-(ps2/ps1)
      St(y,x)
      0
      y1
      ECON377/477 Topic 5.2
      11
    • Economic-theoretic approach: output price indices
      The output price index can also be defined using period-s technology leading to
      These 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 calculated
      Under what conditions are the indices independent of these two factors?
      ECON377/477 Topic 5.2
      12
    • Economic-theoretic approach: output price indices
      These indices are independent of x if and only if the technology is output-homothetic
      A 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 x
      In simple terms, the production possibility curves for different input vectors, x, are all parallel shifts of the production possibility curve for the unit-input vector
      ECON377/477 Topic 5.2
      13
    • Economic-theoretic approach: output price indices
      In 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 derivation
      The 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 properties
      ECON377/477 Topic 5.2
      14
    • Economic-theoretic approach: output price indices
      Since 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 5.2
      15
    • Economic-theoretic approach: output price indices
      We can get close to the above theoretically defined index numbers in equations in a number of ways
      Under 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 indices
      A reasonable approximation to the geometric mean of the two indices is provided by the Fisher output price index number
      ECON377/477 Topic 5.2
      16
    • Economic-theoretic approach: output price indices
      An 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 function
      The translog revenue function is given by
      ECON377/477 Topic 5.2
      17
    • Economic-theoretic approach: output price indices
      We 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 index
      ECON377/477 Topic 5.2
      18
    • Economic-theoretic approach: output price indices
      The 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 data
      Knowledge of the parameters of the translog functions is therefore unnecessary
      ECON377/477 Topic 5.2
      19
    • Economic-theoretic approach: output price indices
      The Törnqvist index is considered to be exact for the translog revenue function
      Also, it is considered superlative since the translog function is a flexible functional form
      That is, it provides a second-order approximation to any arbitrary function
      The Fisher index is exact for a quadratic function and, hence, is also superlative
      ECON377/477 Topic 5.2
      20
    • Economic-theoretic approach: input price indices
      We can measure input price index numbers by comparing costs of producing a vector of outputs, given different input price vectors
      We 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, w
      ECON377/477 Topic 5.2
      21
    • Economic-theoretic approach: input price indices
      We can use the cost function to define input price index numbers
      Given 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 by
      ECON377/477 Topic 5.2
      22
    • Economic-theoretic approach: input price indices
      The cost elements in the equation on the previous slide can be seen from the diagram on the next slide
      The isoquant under technology, Ss, for a given output level, q, is represented by Isoq(q)-Ss
      The 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 5.2
      23
    • Economic-theoretic approach: input price indices
      x2
      b
      a
      B
      x**
      A
      x*
      Isoq(q)-Ss
      0
      a
      b
      A
      B
      x1
      ECON377/477 Topic 5.2
      24
    • Economic-theoretic approach: input price indices
      These points are obtained by shifting lines AA and BB to aa and bb, respectively, where they are tangential to Isoq(y)-Ss
      The 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 5.2
      25
    • Economic-theoretic approach: input price indices
      To compute the input price index, we need to specify the technology and also the output level, q, at which we wish to compute the index
      First, the price index is independent of which period technology we use if and only if the technology exhibits implicit Hicks input neutrality
      Second, 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 homotheticity
      MRS between any inputs is independent of the technology
      ECON377/477 Topic 5.2
      26
    • Economic-theoretic approach: input price indices
      If 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, q
      Two natural specifications are to use the period-s and period-t technologies, along with the output vectors, qs and qt
      ECON377/477 Topic 5.2
      27
    • Economic-theoretic approach: input price indices
      They 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 5.2
      28
    • Economic-theoretic approach: input price indices
      The Laspeyres and Paasche indices provide upper and lower bounds to the economic-theoretic index numbers in the equations on the previous slide
      The geometric mean of these indices can be approximated by the Fisher price index numbers for input prices
      Assume 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 periods
      ECON377/477 Topic 5.2
      29
    • Economic-theoretic approach: input price indices
      Under 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 quantities
      That is,
      where snt and sns are the input expenditure shares of n-th input in periods t and s, respectively
      Törnquist index
      ECON377/477 Topic 5.2
      30
    • Economic-theoretic approach: input price indices
      These 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 use
      They 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 5.2
      31
    • Economic-theoretic approach: input price indices
      The 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 superlative
      The Fisher input price index is exact for a quadratic cost function
      ECON377/477 Topic 5.2
      32
    • Economic-theoretic approach: output quantity indices
      Unlike the case of price index numbers, three strategies can be followed in deriving theoretically sound quantity index numbers
      We focus on only one, the Malmquist index, which is defined using the distance function
      The Malmquist output index, based on technology in period-t, is defined as:
      for an arbitrarily selected input vector, x
      ECON377/477 Topic 5.2
      33
    • Economic-theoretic approach: output quantity indices
      A similar Malmquist index can be defined using period-s technology
      The index defined on the previous slide is independent of the technology involved if and only if the technology exhibits Hicks output neutrality
      The quantity index is independent of the input level, x, if and only if the technology is output homothetic
      MRT between any inputs is independent of the technology
      ECON377/477 Topic 5.2
      34
    • Economic-theoretic approach: output quantity indices
      Even 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 involved
      The 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 form
      ECON377/477 Topic 5.2
      35
    • Economic-theoretic approach: output quantity indices
      Consider output quantity indices based on technology in periods s and t, along with the inputs used in these periods
      We 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 numbers
      A result of particular interest is that the Fisher index provides an approximation to the geometric average of these two indices
      ECON377/477 Topic 5.2
      36
    • Economic-theoretic approach: output quantity indices
      The 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 index
      If 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 5.2
      37
    • Economic-theoretic approach: output quantity indices
      That 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 function
      Since the translog functional form is flexible, the Törnqvist index is also considered to be superlative
      ECON377/477 Topic 5.2
      38
    • Economic-theoretic approach: input quantity indices
      We describe the input quantity index number, derived using the Malmquist distance measure
      Using the concept of the input distance function, we can now define the input quantity index along the same lines as the output index
      We 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 technology
      ECON377/477 Topic 5.2
      39
    • Economic-theoretic approach: input quantity indices
      The 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 5.2
      40
    • Economic-theoretic approach: input quantity indices
      Following 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 technology
      If 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 homotheticity
      ECON377/477 Topic 5.2
      41
    • Economic-theoretic approach: input quantity indices
      Our main purpose is to relate this Malmquist input quantity index number to the input index number derived using some of the formulae above
      The input index defined using base period-s technology is bounded from above by the Laspeyres quantity index
      The index defined on current period-t technology, is bounded from below by the Paasche quantity index
      ECON377/477 Topic 5.2
      42
    • Economic-theoretic approach: input quantity indices
      Therefore, 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 indices
      ECON377/477 Topic 5.2
      43
    • Economic-theoretic approach: input quantity indices
      If 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 equals
      ECON377/477 Topic 5.2
      44
    • Economic-theoretic approach: input quantity indices
      In this equation, snt and sns are input cost-shares in periods t and s, respectively
      This 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 s
      Refer to CROB for a simple numerical example involving index numbers on pages 113-115
      ECON377/477 Topic 5.2
      45
    • Economic-theoretic approach: input quantity indices
      It is easy to see how the index number literature is closely connected with productivity measurement
      For 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 section
      Similarly, 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 numbers
      ECON377/477 Topic 5.2
      46