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  • 1. ECON 377/477
  • 2. 2
    Topic 2
    Productivity and Efficiency Measurement Concepts
  • 3. Productivity and efficiency measurement: concepts
    Concepts and terminology
    Brief description of the methods
    • Data envelopment analysis
    • 4. Index numbers
    • 5. Stochastic frontier analysis
    Econ 377/477 Topic 2
    3
  • 6. Terminology
    Productivity
    Technical efficiency
    Allocative efficiency
    Cost efficiency
    Technical change
    Scale efficiency
    Constant returns to scale ...
    Econ 377/477 Topic 2
    4
  • 7. Terminology
    • Point of (technically) optimal scale
    • 8. Total factor productivity (TFP)
    • 9. Production frontier
    • 10. Feasible production set
    • 11. Output-mix allocative efficiency
    • 12. Revenue efficiency
    5
    Econ 377/477 Topic 2
  • 13. Set theoretic representation of a production technology
    • Production function: a single-output technology
    • 14. Production technology: a multiple-output production process
    • 15. The technology set consists of all input-output vectors (x,q) such that x can produce q:
    S = {(x,q): x can produce q}
    Properties: CROB, pp 43-44
    Econ 377/477 Topic 2
    6
  • 16. Set theoretic representation of a production technology
    • Output sets: P(x) = {q: x can produce q}
    = {q: (x,q)  S}
    • Input sets: L(q) = {x: x can produce q}
    = {x: (x,q)  S}
    • Note the properties of the output set defined by CROB
    Properties: CROB, pp 43-44
    7
    Econ 377/477 Topic 2
  • 17. Production possibility curves and revenue maximisation: a digression
    • Consider a one-input two-output example, and specify an input requirement function:
    x1 = g(q1,q2)
    to illustrate a production possibility curve (PPC)
    • The isorevenue line is the negative ratio of the output prices
    • 18. The optimal (revenue-maximising) point is the point of tangency between this line and the PPC, shown in the diagram on the next slide
    Econ 377/477 Topic 2
    8
  • 19. Production possibility curves and revenue maximisation: a digression
    q2
    Optimal point
    A
    Isorevenue line (slope = -p1/p2)
    PPC(x1=x10)
    0
    q1
    Econ 377/477 Topic 2
    9
  • 20. Technical change and the PPC
    • Technical change can favour the production of one commodity over another, illustrated on the next slide
    • 21. The outward shift of the PPC in part (a) is consistent across the output-output space
    • 22. In contrast, the outward shift of the PPC in part (b) is greater close to the horizontal axis than it is close to the vertical axis
    • 23. That is, technical change favours the production of q1 over q2
    10
    Econ 377/477 Topic 2
  • 24. Technical change and the PPC
    q2
    q2
    PPC(x=x10,t=1)
    PPC(x=x10,t=1)
    PPC(x=x10,t=0)
    PPC(x=x10,t=0)
    0
    q1
    q1
    0
    Neutral technical change
    Non-neutral technical change
    Econ 377/477 Topic 2
    11
  • 25. Output and input distance functions
    Distance functions are closely related to production frontiers
    Radial contractions and expansions are involved in defining distance functions
    They enable description of multi-input and multi-output production technology without the need to specify a behavioural objective
    Either output or input distance functions are usually specified, but there are also directional distance functions
    Econ 377/477 Topic 2
    12
  • 26. Output distance functions
    The diagram on the next slide demonstrates an output distance function
    An output distance function considers a maximal proportional expansion of the output vector, given an input vector
    It is defined on the output set, P(x), as:
    do(x,q) = min{δ: (q/δ)P(x)}
    Econ 377/477 Topic 2
    13
  • 27. Output distance functions
    q2
    • = 0A/0B
    • 28. The reciprocal is the factor by which the production of all output quantities can be increased while remaining within the feasible PPC for the given input level
    • 29. B and C would have distance function values of 1
    • 30. A would have a distance function value that is less than 1
    B
    A
    q2A
    C
    PPC-P(x)
    P(x)
    0
    q1
    q1A
    14
    Econ 377/477 Topic 2
  • 31. Input distance functions
    The diagram on the next slide demonstrates an input distance function
    An input distance function characterises the production technology as the minimal proportional contraction of the input vector, given an output vector
    It is defined on the input set, L(q), as:
    di(x,q) = max{ρ: (x/ρ)L(q)}
    Econ 377/477 Topic 2
    15
  • 32. Input distance functions
    The function di(x,q) = max{:x/ )L(q)} is:
    • non-decreasing in x
    • 33. non-increasing in q
    • 34. linearly homogeneous in x
    • 35. quasi-concave in q and concave in x
    If x  L(q), then di (x,q)  1; ρ = 0A/0B
    If both inputs and outputs are weakly disposable, then di(x,q)  1 iffdo(x,q)  1
    Under constant returns to scale:
    di (x,q) = 1/do(x,q) for all x and q
    X2
    A
    X2A
    B
    C
    L(q)
    Isoq-L(q)
    0
    X1
    X1A
    16
    Econ 377/477 Topic 2
  • 36. Efficiency measurement concepts
    References: Farrell (1957) and Debreu (1951)
    Two efficiency components are:
    • Technical efficiency
    the ability of the firm to obtain maximal output from given sets of inputs
    • Allocative efficiency
    the ability of the firm to use the inputs in optimal proportions, given their respective prices
    Economic efficiency (EE) is the product of technical efficiency (TE) and allocative efficiency (AE)
    Econ 377/477 Topic 2
    17
  • 37. Input-orientated measures
    Assume two inputs (x1and x2) and one output (q) under the assumption of constant returns to scale
    The unit isoquant of fully efficient firms is represented by SS’
    Suppose a firm uses quantities of inputs defined by point P in the graphical representation on the next slide
    Its technical efficiency is represented by the distance, QP
    Econ 377/477 Topic 2
    18
  • 38. Input-orientated measures
    X2/q
    S
    P
    A
    Q
    R
    Q’
    S’
    0
    X1/q
    A’
    19
    Econ 377/477 Topic 2
  • 39. Interpreting input-orientated measures
    Technical inefficiency can be represented as the distance QP – the amount of inputs that could be proportionally reduced without reducing output
    It represents the percentage by which all inputs need to be reduced to achieve technically efficient production, the ratio QP/0P
    The technical efficiency index, 0Q/0P, lies between zero and one
    A firm is fully efficient if it has a technical efficiency index of 1
    Econ 377/477 Topic 2
    20
  • 40. Interpreting input-orientated measures
    Given an input-price ratio, the allocative efficiency index is 0R/0Q
    No reduction in cost is possible if production is at Q’
    Point Q is technically efficient but allocatively inefficient
    Econ 377/477 Topic 2
    21
  • 41. Interpreting input-orientated measures
    Economic efficiency is EEi= 0R/0P
    That is, EE = TE*AE
    We assume the production function of the fully efficient firm is known
    Note that this is not always the case!
    An efficient isoquant must be estimated using sample data
    22
    Econ 377/477 Topic 2
  • 42. Output-orientated measures
    To specify a parametric frontier production function in input-output space, consider a Cobb-Douglas function:
    ln(yi) = f(ln(xi), ) - ui
    where:
    yi is the output of the i-th firm
    xi is an input vector
    ui is a non-negative variable representing inefficiency
    23
    Econ 377/477 Topic 2
  • 43. Output-orientated measures
    Technical efficiency is calculated as:
    TEi = yi/f(ln(xi),) = exp (-ui)
    An output-orientated measure indicates the magnitude of the output of the i-th firm relative to the output that could be produced by the fully efficient firm using the same input vector
    This approach does not account for noise and we need to impose a functional form
    Econ 377/477 Topic 2
    24
  • 44. Output-orientated measures
    The difference between input- and output-orientated measures can be illustrated by a simple example of one input, x, and one output, q
    The first diagram on the next slide shows the case of a decreasing-returns-to-scale technology, f(x), and an inefficient firm operating at point P
    The Farrell input-orientated measure of TE is the ratio, AB/AP, and the output-orientated measure of TE is the ratio, CP/CD
    They are only equal when there are constant returns to scale, in the second part of the diagram
    Econ 377/477 Topic 2
    25
  • 45. Output-orientated measures and returns to scale
    q2
    (a) DRS
    (b) CRS
    f(x)
    f(x)
    D
    D
    B
    B
    A
    A
    P
    P
    0
    0
    C
    C
    26
    Econ 377/477 Topic 2
  • 46. Output-orientated measures
    An output-orientated measure of TE with two outputs, q1 and q2, is shown on the next slide
    Assuming CRS, we can represent the technology by a unit PPF, ZZ’, in two dimensions
    A corresponds to an inefficient firm
    Output-orientated technical efficiency is the ratio:
    TE = 0A/0B, = do(x,q)
    where do(x,q) is the output distance function at the observed output vector of the firm associated with point A
    Econ 377/477 Topic 2
    27
  • 47. Efficiency measures with an output orientation
    q2/x1
    D
    TEo=OA/OB
    C
    B
    AEo=OB/OC
    A
    B’
    REo=OA/OC
    D’
    0
    q1/x1
    28
    Econ 377/477 Topic 2
  • 48. Output-orientated measures
    Allocative efficiency (AE) is equal to 0B/0C
    Revenue efficiency (RE) is equivalent to economic efficiency
    It is defined for an output price vector p represented by the line, DD’
    RE = 0A/0C, = p’q/p’q*
    where q* is the revenue-efficient vector associated with the point B’
    Econ 377/477 Topic 2
    29
  • 49. A few points
    TE has been measured along a ray from the origin to the observed production point
    All points measured along the ray from the origin to the observed production point hold relative proportions of inputs (outputs) constant
    Changing the units of measurement will not change the value of the efficiency measure
    Econ 377/477 Topic 2
    30
  • 50. A few points
    We have demonstrated measures of allocative efficiency from cost-minimising and revenue-maximising perspectives
    We can also do the same for the profit-maximising perspective, using DEA and stochastic frontier analysis (SFA)
    For SFA, profit efficiency is decomposed using:
    input-allocative efficiency
    output-allocative efficiency
    input-orientated technical efficiency
    31
    Econ 377/477 Topic 2
  • 51. Measuring productivity and productivity change
    Econ 377/477 Topic 2
    32
    Measuring productivity of a firm and change in productivity is part of performance measurement
    Partial measures were discussed in Topic 1
    We represent change (growth and decrease) of productivity by a total factor productivity (TFP) index, also known as a multifactor productivity index (MFP)
  • 52. Measuring productivity and productivity change
    Consider the problem of measuring productivity change for a firm from period s to period t
    The typical period used is a year
    Assume that the firm makes use of the state of knowledge, as represented by production technologies Ss and St in periods s and t
    Econ 377/477 Topic 2
    33
  • 53. Measuring productivity and productivity change
    Suppose the firm produces outputs qsand qtusing inputs xsand xt, respectively
    In some cases, we may have information on output and input prices, represented by output price vectors psand pt, andinput vectors, wsand wt,in periods s and t, respectively
    34
    Econ 377/477 Topic 2
  • 54. Measuring productivity and productivity change
    Given these data on this firm, how do we measure productivity change?
    There are several simple and intuitive approaches we can use to derive meaningful measures of productivity change
    We consider four possible alternatives, outlined in the following slides
    Econ 377/477 Topic 2
    35
  • 55. Approaches to measure productivity change
    Hicks-Moorsteen approach
    This approach simply uses a measure of productivity as output growth net of growth in inputs
    If output has doubled from period s to period t,and if this output growth was achieved using only a 60 per cent growth in input use, we conclude that the firm has achieved productivity growth
    Econ 377/477 Topic 2
    36
  • 56. Approaches to measure productivity change
    Hicks-Moorsteen approach (continued)
    It is easy to measure and interpret, but quite difficult to identify the main sources of productivity growth
    Suppose productivity has grown by 10 per cent; do we attribute this to technical change or to improvements in efficiency?
    Econ 377/477 Topic 2
    37
  • 57. Approaches to measure productivity change
    Profitability approach
    This approach measures productivity change using growth in profitability after making appropriate adjustments for movements in input prices and output prices from period s to period t
    Econ 377/477 Topic 2
    38
  • 58. Approaches to measure productivity change
    Profitability approach (continued)
    Since the TFP measure in equation (3.21) in the text book does not contain any price effects, the main sources of TFP change over periods s and t can be attributed to technical change and (technical, allocative and scale) efficiency changes over this period
    Econ 377/477 Topic 2
    39
  • 59. Approaches to measure productivity change
    Econ 377/477 Topic 2
    40
    CCD Approach
    This approach measures productivity by comparing the observed outputs in period s and period t with the maximum level of outputs (keeping the output mix constant)
    A commonly used index that follows this approach is the Malmquist index
  • 60. Malmquist TFP index: issues
    Refer to CROB, pages 69-74, for discussion of the following issues:
    • Malmquist TFP and orientation
    • 61. Malmquist and HM TFP indices
    • 62. Malmquist TFP and technical inefficiency
    • 63. Malmquist TFP and returns to scale
    • 64. Malmquist TFP and transitivity
    Econ 377/477 Topic 2
    41
  • 65. Malmquist output-orientated TFP index
    Output-orientated measures ofproductivity focus on the maximum level of outputs that could be produced using a given input vector and a given production technology relative to the observed level of outputs:
    Assuming technical efficiency in both periods:
    Econ 377/477 Topic 2
    42
  • 66. Malmquist input-orientated TFP Index
    The input-orientated productivity focuses on the level of inputs necessary to produce observed output vectors qsand qt under a reference technology:
    Assuming technical efficiency in both periods:
    Econ 377/477 Topic 2
    43
  • 67. Approaches to measure productivity change
    Econ 377/477 Topic 2
    44
    Component-based approach
    Following this approach, various sources of productivity growth are identified: technical change; efficiency change; change in the scale of operations; and output mix effects
    If we can measure these effects separately, then productivity change can then be measured as the product (or sum total) of all these individual effects
  • 68. TFP Index: measurement by sources of productivity change
    Technical change
    Technical efficiency change
    Scale efficiency change
    Optimal mix effect
    TFP change = technical change  TE change  scale efficiency change  output mix effect
    Econ 377/477 Topic 2
    45