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
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
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:
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:
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
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