Abstract
To explain variations in managerial performance, we need to know something about what can and cannot be produced using different production technologies. In this book, a production technology (or simply ‘technology’) is defined as a technique, method or system for transforming inputs into outputs. For most practical purposes, it is convenient to think of a technology as a book of instructions, or recipe. In this book, the set of technologies that exist in a given period is referred to as a technology set. If we think of a technology as a book of instructions, then we can think of a technology set as a library. The input-output combinations that are possible using different technologies can be represented by output sets, input sets and production possibilities sets. Under certain conditions, they can also be represented by distance, revenue, cost and profit functions. This chapter defines, and discusses the properties of, these different sets and functions.
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Notes
- 1.
Coelli et al. (2005, p. 192) refer to strongly disposable environmental variables as ‘positive effect’ environmental variables.
- 2.
The reference vectors in O11 and O13 are arbitrary. Vectors of ones have been chosen here for notational convenience. The choice of reference period in O13 is also arbitrary. Again, period 1 has been chosen for notational convenience.
- 3.
In I12 and I14, the reference vectors are again arbitrary. In I14, the choice of reference period is also arbitrary.
- 4.
If O1 and O2 are true, then DO1 and DO2 are true. The first part of DO3 is obvious. The last two parts of DO3 are equivalent to D\(_o\).1 and D\(_o\).5 in Färe and Primont (1995, pp. 17, 18) . DO4 \(\Leftrightarrow \) \(\nabla .7\) in Shephard (1970 , pp. 208, 211) which is satisfied under D\(_o\).4 in Färe and Primont (1995) . DO2 and O4 \(\Rightarrow \) DO5. DO5 is equivalent to D\(_o\).4 in Färe and Primont (1995).
- 5.
O6s \(\Rightarrow \) DO6 and O7s \(\Rightarrow \) DO7 (Shephard 1970 , proof of \(\nabla .5\) and \(\nabla .8\) on pp. 210–211). If O1, O2, O4, O5, O6s, O7s, O9s, O15 and I16 are true, then \(\overline{\text {A.1}}\)–\(\overline{\text {A.8}}\) in Shephard (1970) are true. Then DO8 and DO9 follow from Shephard (1970, pp. 207, 208, Prop. 61).
- 6.
Proofs of DO11 and DO13 are given in Appendix A.1 (Propositions 1 and 3).
- 7.
A proof is given in Appendix A.1 (Proposition 5).
- 8.
A proof is given in Appendix A.1 (Proposition 17).
- 9.
I6s \(\Rightarrow \) DI6 and I7s \(\Rightarrow \) DI7 (Shephard 1970 , proof of D.4, D.5 and D.8 on pp. 68–70). If I1, I2, I4, I5, I6s, I7s, I9s, I15 and I16 are true, then \(\overline{\text {A.1}}\)–\(\overline{\text {A.8}}\) in Shephard (1970) are true. Then DI4–DI9 follow from Shephard (1970, pp. 207, Proposition 60)
- 10.
Proofs of DI12 and DI14 are given in Appendix A.1 (Propositions 9 and 11).
- 11.
A proof is given in Appendix A.1 (Proposition 13).
- 12.
A proof is given in Appendix A.1 (Proposition 18).
- 13.
The term ‘technical’ is used here to distinguish the MRTS from a similar concept in consumer demand theory. In consumer demand theory, the marginal rate of substitution (MRS) is the rate at which consumers can exchange one good for another while holding utility and all other variables fixed.
- 14.
Proofs of R11 and R13 are given in Appendix A.1 (Propositions 2 and 4).
- 15.
A proof is given in Appendix A.1 (Proposition 6).
- 16.
Proofs of C12 and C14 are given in Appendix A.1 (Propositions 10 and 12).
- 17.
A proof is given in Appendix A.1 (Proposition 14).
- 18.
A set is compact if it is closed and bounded. If the set of technically-feasible output-input combinations that yield nonnegative profit is compact, then profit achieves a maximum on \(T^t(z)\). This means the maximum operator can be used in (2.25) instead of the supremum operator.
- 19.
- 20.
For proofs of F7s and F15, see Shephard (1970, p. 21) . Proofs of F10, F11 and F13 are left as an exercise for the reader.
- 21.
In this example, changes in A(t) are attributed to the discovery of new technologies. In this book, such changes are referred to as technical change. Solow (1957) also attributes changes A(t) to technical change. However, he “[uses] the phrase ‘technical change’ as a shorthand expression for any kind of shift in the production function. Thus, speedups, improvements in the education of the labor force, and all sorts of things will appear as ‘technical change”’ (Solow 1957, p. 312).
- 22.
For proofs of H6s and H16, see Shephard (1970, p. 198) . Proofs of H10, H12 and H14 are left as an exercise for the reader.
- 23.
T1–T7 \(\Rightarrow \) GR.1–GR.5 in Färe et al. (1985, p. 111) . DH2–DH5 are equivalent to \(F_g.1\), \(F_g.3\), \(F_g.4\) and \(F_g.5\) in Färe et al. (1985, pp. 111, 112) (respectively). A function F(x, y) is said to be almost homogeneous of degree a, b, and c, respectively, if and only if \(F(\lambda ^ax,\lambda ^by)=\lambda ^cF(x,y)\) for any \(\lambda >0\) (Lau 1972, p. 282).
- 24.
- 25.
- 26.
Humphrey and Pulley (1997) refer to their nonstandard profit function as an ‘alternative indirect profit function’.
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O’Donnell, C.J. (2018). Production Technologies. In: Productivity and Efficiency Analysis. Springer, Singapore. https://doi.org/10.1007/978-981-13-2984-5_2
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