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AEB 6184 – Review and Frontier Models

AEB 6184 – Review and Frontier Models. Elluminate - 10. Review of the Production Function. Kumbhakar , S.C. and C.A. Knox Lovell. 2000. Stochastic Frontier Analysis .

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AEB 6184 – Review and Frontier Models

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  1. AEB 6184 – Review and Frontier Models Elluminate - 10

  2. Review of the Production Function • Kumbhakar, S.C. and C.A. Knox Lovell. 2000. Stochastic Frontier Analysis. • Definition 2.1: The graph of the production technology, GR={(y,x): x can produce y}, describes the set of feasible input-output vectors. [pp.18-19]. • Definition 2.2: The input sets of production technology, L(y)={x:(y,x)  GR}, describe the sets of input vectors that are feasible for each output y R+M.[p. 21 see Shephard p 13 – Level sets]. • Definition 2.3: The output sets of production technology P(x) = {y:(y,x)  GR}, describe the sets of output vectors that are feasible for each input vector x  R+n. [p. 22]

  3. Definition 2.4: The input isoquantsIsoL(y) = {x: x L(y), λx  L(x), λ < 1} describe the sets of input vectors capable of producing each output vector y but which, when radially contracted, became incapable of producing output vector y. [p. 23] • Definition 2.5: The input efficiency subsetsEffL(y) = {x: x L(y), x’ ≤ x→ x’  L(y)} describe the sets of input vectors capable of producing each output vector y but which, when contracted in any dimension, become incapable of producing the output vector y. [p.23] • Definition 2.9: A production frontier is a function f(x) = max[y: y P(x)} = max[y: x  L(y)]. [p. 26, see Shephard pp. 20-23].

  4. Bridging to New Materials • We will be developing the dual relationships using three different approaches: • Diewert – Generalized Leontief • Shephard – Distance Functions • Lau – Conjugate Duality • A first step is to define the distance functions – Starting with Kumbhakar and Lovell. • Definition 2.10: An input distance function is a function DI(y,x) = max[λ: x/λ L(y)]. [p. 28]

  5. Input Distance Functions x/λ x/λ

  6. Distance Functions • From Kumbhakar and Lovell’s perspective, the value of the distance function is a measure of inefficiency. • If λ>1 then the current vector is inefficient. • From Shephard’s viewpoint, λ defines the efficient frontier. • As noted in Kumbhakar and Lovell • Definition 2.12: A cost function is a function c(y,w) = minx[w’x: x L(y)] = minx[w’x: DI(y,x)≥1]. [p.33]. • We will start developing this dual function in more detail in future lectures.

  7. Cost Function • The dominant theme in duality is that under some assumptions cost minimizing or profit maximizing behavior retains the information in the production function. • We start with a simple quadratic cost function

  8. Optimality • First we define optimality from the level of output • Next, looking ahead, we define the optimal levels of input using Shephard’s lemma

  9. Sample Draw (Without Error)

  10. Least Squares Estimates of Production Function • Representing the technology as a quadratic production function • Estimates

  11. Problem Set

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