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  • 1. ECON 377/477
  • 2. Topic 5.4
    Measuring TFP change
  • 3. Outline
    Introduction
    Malmquist TFP index and panel data
    Calculating TFP change using DEA frontiers
    Calculating TFP change using SFA frontiers
    Measuring TFP change: conclusions
    Summary of methods
    3
    ECON377/477 Topic 5.4
  • 4. Introduction
    In this part, we consider the case where we have access to better data
    In particular, we assume we have data on a sample of firms in periods s and t that are sufficient to obtain an estimate of the production technology in these two periods
    We can then calculate the required distances directly and relax a number of the assumptions made to construct the TFP index numbers
    In particular, we need no longer assume all firms are operating on the surface of the production technology (are technically efficient)
    ECON377/477 Topic 5.4
    4
  • 5. Introduction
    We no longer have the situation where the ratio of the distance functions provides a measure of TFP change that is identical to technical change (TC) (i.e. frontier shift), as was the case in earlier parts of this topic
    Thus, when panel data are available, we can obtain a measure of TFP change that has two components:
    a TC component
    a TE change component
    ECON377/477 Topic 5.4
    5
  • 6. Introduction
    The remainder of this part is organised as follows
    The Malmquist TFP index is briefly described
    Then, we describe how to calculate these indices using DEA-like methods
    Next, we describe the calculation of these indices using the SFA methods
    We then refer to detailed application of some of these methods in CROB
    Finally, we make some brief concluding comments
    ECON377/477 Topic 5.4
    6
  • 7. Malmquist TFP index and panel data
    Malmquist TFP change measures can be decomposed into various components, including TC and TE change
    These measures could be calculated using distances measured relative to DEA frontiers
    The Malmquist TFP index measures the TFP change between two data points by calculating the ratio of the distances of each data point relative to a common technology
    ECON377/477 Topic 5.4
    7
  • 8. Malmquist TFP index and panel data
    If the period t technology is used as the reference technology, the Malmquist (output-orientated) TFP change index between period s and period t is defined as:
    Alternatively, if the period s reference technology is used, it is defined as:
    ECON377/477 Topic 5.4
    8
  • 9. Malmquist TFP index and panel data
    The notation represents the distance from the period t observation to the period s technology
    A value of mo greater than one indicates positive TFP growth from period s to period t while a value less than one indicates a TFP decline
    These period s and period t indices are only equivalent if the technology is Hicks output-neutral
    It is Hicks output-neutral if the output distance functions may be represented as
    for all t
    ECON377/477 Topic 5.4
    9
  • 10. Malmquist TFP index and panel data
    The Malmquist TFP index is often defined as the geometric mean of the two indices with period s and period t technology:
    The distance functions can be rearranged to show that it equates to the product of a technical efficiency change index and an index of TC:
    TE change
    TC
    ECON377/477 Topic 5.4
    10
  • 11. Malmquist TFP index and panel data
    The ratio outside the square brackets in the second equation on the previous slide measures the change in the output-oriented measure of Farrell technical efficiency between periods s and t
    The remaining part of the index in this equation is a measure of TC: the geometric mean of the shift in technology between the two periods, evaluated at xt and xs
    ECON377/477 Topic 5.4
    11
  • 12. Malmquist TFP index and panel data
    A number of additional possible decompositions of these technical efficiency change and TC components have been proposed, notably:
    • decomposing TC into input bias, output bias and ‘magnitude’ components
    • 13. decomposing TE change into scale efficiency and ‘pure’ technical efficiency components (this can only be done when the distance functions are estimated relative to a CRS technology
    ECON377/477 Topic 5.4
    12
  • 14. Malmquist TFP index and panel data
    The decomposition into scale efficiency involves taking the efficiency change measure and decomposing it into a pure efficiency change component (measured relative to the arguably true VRS frontier),
    and a scale efficiency component,
    ECON377/477 Topic 5.4
    13
  • 15. Malmquist TFP index and panel data
    The scale efficiency change component is the geometric mean of two scale efficiency change measures
    The first is relative to the period t technology and the second is relative to the period s technology
    Note that the extra subscripts, v and c, relate to the VRS and CRS technologies, respectively
    Refer to CROB for an evaluation of this method and an alternative approach
    The returns to scale properties of the technology are very important in TFP measurement
    ECON377/477 Topic 5.4
    14
  • 16. Calculating TFP change using DEA frontiers
    A number of different methods could be used to estimate a production technology and, hence, measure the distance functions that make up the Malmquist TFP index
    The most popular method has been the DEA-like LP method, discussed in this section
    The other main alternative approach is the use of stochastic frontier methods, described in the next section
    ECON377/477 Topic 5.4
    15
  • 17. Calculating TFP change using DEA frontiers
    Given suitable panel data, we can calculate the distance measures using DEA-like LPs
    For the i-th firm, four distance functions are calculated to measure the TFP change between two periods
    This requires solving four LP problems using a CRS technology in the TFP calculations
    This ensures that TFP change measures satisfy the property that if all inputs are multiplied by the (positive) scalar  and all outputs are multiplied by the (non-negative) scalar , the resulting TFP change index will equal /
    ECON377/477 Topic 5.4
    16
  • 18. Calculating TFP change using DEA frontiers
    The required four LPs are:
    [dot(qt, xt)] - 1 = max,
    st -qit + Qt 0
    xit - Xt0
    0
    [dos(qs, xs)] - 1 = max,
    st -qis+ Qs0
    xis – Xs0
    0

    ECON377/477 Topic 5.4
    17
  • 19. Calculating TFP change using DEA frontiers
    [dot(qs, xt)] - 1 = max,
    st -qis + Qt 0
    xis - Xt 0
     0
    [dos(qt, xt)] - 1 = max,
    st -qit + Qs 0
    xit – Xs 0
     0
    ECON377/477 Topic 5.4
    18
  • 20. Calculating TFP change using DEA frontiers
    In the third and fourth LPs, where production points are compared with technologies from different time periods, the  parameter need not be greater than or equal to one, as it must be when calculating Farrell output-orientated TEs
    The s and s are likely to take different values in the above four LPs, which must be solved for each firm in the sample
    As extra time periods are added, it is necessary to solve an extra three LPs for each firm (to construct a chained index)
    ECON377/477 Topic 5.4
    19
  • 21. Calculating TFP change using DEA frontiers
    Decomposing technical efficiency change into scale efficiency and ‘pure’ technical efficiency measure requires the solution of two additional LPs (when comparing two production points)
    These LPs would involve repeating the first two with the convexity restriction (I1 = 1) added to each
    This provides estimates of distance functions relative to a VRS technology
    ECON377/477 Topic 5.4
    20
  • 22. Calculating TFP change using DEA frontiers
    CROB (pages 295-300) provide a numerical example of the application of the DEA-like method to construct Malmquist TFP indices, using the DEAP computer program
    Four distances are calculated for each firm in each year, relative to:
    the previous period’s CRS DEA frontier
    the current period’s CRS DEA frontier
    the next period’s CRS DEA frontier
    the current period’s VRS frontier
    ECON377/477 Topic 5.4
    21
  • 23. Calculating TFP change using DEA frontiers
    All indices are calculated relative to the previous year
    Hence, the output begins with year 2
    Five indices are presented for each firm in each year:
    TE change relative to a CRS technology
    technological change
    pure TE change relative to a VRS technology
    scale efficiency change
    TFP change
    ECON377/477 Topic 5.4
    22
  • 24. Calculating TFP change using DEA frontiers
    Summary tables of these indices follow for the different time periods (over all firms) and for the different firms (over all time periods)
    Note that all indices are equal to one for time period 3 because the data for year 3 are identical to the year 2 data in the example data set used
    ECON377/477 Topic 5.4
    23
  • 25. Calculating TFP change using SFA frontiers
    The distance measures required for the Malmquist TFP index calculations can also be measured relative to a parametrically estimated technology
    We focus our attention on the production frontier case, which is a single-output special case of the more general (multi-output) output distance function
    ECON377/477 Topic 5.4
    24
  • 26. Calculating TFP change using SFA frontiers
    Consider a translog stochastic production frontier defined as:
    i = 1,2,...,I , t = 1,2,...,T
    where qit is the output of the i-th firm in the t-th year; xnit denotes the n-th input variable; t is a time trend representing TE; the s are unknown parameters to be estimated; the vits are random errors, and the uits are the technical inefficiency effects
    ECON377/477 Topic 5.4
    25
  • 27. Calculating TFP change using SFA frontiers
    The above model has the time trend, t, interacted with the input variables to allow for non-neutral TC
    The technical efficiencies of each firm in each year can be predicted using the approach outlined in Topic 4
    We obtain the conditional expectation of exp(-uit), given the value of eit= vit - uit
    These TE predictions are between zero and one, with a value of one indicating full TE
    ECON377/477 Topic 5.4
    26
  • 28. Calculating TFP change using SFA frontiers
    We can use measures of TE and TC to calculate the Malmquist TFP index via equations (11.4-11.6) in CROB (see next slide)
    TE change = TEit/TEis
    The TC index between period s and t for the i-th firm can be calculated directly from the estimated parameters
    First, the partial derivatives of the production function are evaluated with respect to time using the data for the i-th firm in periods s and t
    ECON377/477 Topic 5.4
    27
  • 29. Malmquist TFP index and panel data
    From an earlier slide (slide 9), the Malmquist output index equates to the product of a technical efficiency change index and an index of TC:
    TE change
    TC
    CROB, equation (11.4)
    ECON377/477 Topic 5.4
    28
  • 30. Calculating TFP change using SFA frontiers
    The TC index between the adjacent periods s and t is calculated as the geometric mean of these two partial derivatives
    When a translog function is involved, this is equivalent to the exponential of the arithmetic mean of the log derivatives, and TC equals:
    The indices of TE change and TC can then be multiplied together to obtain a Malmquist TFP index
    ECON377/477 Topic 5.4
    29
  • 31. Calculating TFP change using SFA frontiers
    The above technical change measure involves derivative calculations, which appears to contradict the earlier comments that these indices are derived from distance measures
    It can be easily shown (for the translog case in which a time trend is used to represent technical change) that the geometric means of the distance ratios in equation (11.6) in CROB are equivalent to the geometric means of the derivative measures
    ECON377/477 Topic 5.4
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  • 32. Calculating TFP change using SFA frontiers
    A criticism of this method is that the TFP index may produce biased measures because the productivity changes due to scale changes are not captured
    One possible solution to this problem is to impose CRS upon the estimated production technology
    Another option is to derive a Malmquist TFP decomposition identical to that proposed above, and address the scale issue by including a scale change component in the TFP measure
    ECON377/477 Topic 5.4
    31
  • 33. Calculating TFP change using SFA frontiers
    Scale change =
    where , and
    This scale change index is equal to one if the production technology is CRS, when the scale elasticity ( ) equals one
    An empirical application to rice production data is provided by CROB (pages 302-309)
    ECON377/477 Topic 5.4
    32
  • 34. Measuring TFP change: conclusions
    Some of the advantages of the frontier approach are:
    • It does not require price information
    • 35. It does not assume all firms are fully efficient
    • 36. It does not need to assume a behavioural objective such as cost minimisation or revenue maximisation
    • 37. It permits TFP change to be decomposed into components such as TC, TE change and scale change
    ECON377/477 Topic 5.4
    33
  • 38. Measuring TFP change: conclusions
    An important advantage of the Tornqvist approach is that it can be calculated using only two data points
    In contrast, the frontier approach needs a number of firms to be observed in each time period so that the frontier technology in each year can be estimated
    If one has suitable panel data, the frontier approach provides richer information and makes fewer assumptions
    ECON377/477 Topic 5.4
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  • 39. Measuring TFP change: conclusions
    But if only aggregate time-series data are available, the Tornqvist approach provides useful estimates of TFP change, given that the assumptions specified earlier in the topic notes are reasonable
    ECON377/477 Topic 5.4
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  • 40. Summary of methods
    We have considered four principal methods:
    Least-squares econometric production models
    TFP indices (Tornqvist/Fisher)
    DEA
    SFA
    They differ in various ways, as demonstrated in the table on the next slide
    ECON377/477 Topic 5.4
    36
  • 41. Summary of methods
    ECON377/477 Topic 5.4
    37
  • 42. Summary of methods
    Efficiency is generally measured using either DEA or SFA
    Advantages of SFA over DEA include:
    • it accounts for noise
    • 43. it can be used to conduct conventional tests of hypotheses
    Disadvantages include:
    • the need to specify a distributional form for the inefficiency term
    • 44. the need to specify a functional form for the production function (or cost function)
    ECON377/477 Topic 5.4
    38
  • 45. Summary of methods
    TC (or TFP) is usually measured using either least squares econometric methods or Tornqvist/Fisher index numbers
    Some of the advantages of index numbers over least-squares econometric methods are:
    • only two observations are needed
    • 46. they are easy to calculate
    • 47. the method does not assume a smooth pattern of TC
    The principal disadvantage is that it requires both price and quantity information
    ECON377/477 Topic 5.4
    39
  • 48. Summary of methods
    Both approaches assume that firms are technically efficient, which is unlikely to be true
    To relax this assumption, frontier methods can be used, assuming panel data are available, to calculate TFP change
    ECON377/477 Topic 5.4
    40
  • 49. Summary of methods
    Some of the advantages of the SFA approach over the Tornqvist/Fisher index numbers approach are that it:
    • does not require price information
    • 50. does not assume all firms are fully efficient
    • 51. does not require the assumption of cost minimisation and revenue maximisation
    • 52. permits TFP to be decomposed into TC and TE change
    ECON377/477 Topic 5.4
    41
  • 53. Summary of methods
    But an important advantage of the index-number approach is that it only requires two data points, say observations on two firms in one time period or observations on one firm in two time periods
    The frontier approaches need a number of firms to be observed in each time period so that the frontier technology in each year can be calculated
    ECON377/477 Topic 5.4
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