Abstract
This contribution aims to detect and measure more severe forms of congestion than the ones that could hitherto be evaluated in axiomatic production theory. To this end, we define a new S-disposal axiom, a kind of limited strong disposability. This S-disposal assumption leads to a duality result between a general input directional distance function and the cost function that is weaker than the ones established in the literature. Finally, we indicate how finite data sets can or cannot be rationalized by a minimal technology compatible with S-disposal, thereby generalizing the nonparametric weak axiom of cost minimization test.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The latter phase is known as the toxic range of nutrients in soil science.
The few empirical studies using this specification focused mainly on disembodied technical change that widens productive factor combinations rather than detecting congestion.
Since the proposed S-disposal assumption limits the extent of strong disposability but does not comply with any local notion in a mathematical sense, we opt for the adjective limited.
Murty et al. (2012) argue against this rather widespread use of ray disposability to model the relation between good and bad outputs. These authors explicitly combine a standard technology with good inputs and outputs with a residual generating technology which does not satisfy standard free disposal axioms. We ignore this particular application area focusing on the relation between good and bad outputs and focus on modeling congestion between good inputs and outputs instead.
Since the acceptance/rejection of convexity may constitute a dividing line between economics and ecology (e.g., Dasgupta and Mähler 2003), it remains to be seen how trade-offs between good and bad outputs can be modeled without convexity.
Fuss et al. (1978, p. 223) state: “Given the qualitative, nonparametric nature of the fundamental axioms, this suggests [ ] that the more relevant tests will be nonparametric, rather than based on parametric functional forms, even very general ones.”
Using negation, one can easily see that this formulation of free disposability is logically equivalent with \(\forall x\in L(y): u\ge x \Rightarrow u\in L(y)\), the latter being somewhat more common.
Kuosmanen (2005) shows that this traditional specification fails convexity, but that a revised specification is convex.
The figure also reminds us about the possibility that there may be one or several bliss points where production is maximal. However, to clearly discern such a case one would need an approach also considering the output dimensions rather than just focusing on the input dimensions alone.
As pointed out by an anonymous referee, in general the efficient subset is not closed (see for instance Arrow et al. 1953).
Also McFadden (1978, 60) anticipates the use of negative prices and maintains that duality results can be preserved under these circumstances.
References
Arrow K, Barankin E, Blackwell D (1953) Admissible points of convex sets. In: Kuhn H, Tucker A (eds) Contributions to the theory of games, vol II. Princeton University Press, Princeton, pp 87–91
Barnett W (2002) Tastes and technology: curvature is not sufficient for regularity. J Econometr 108(1):199–202
Blow L, Browning M, Crawford I (2008) Revealed preference analysis of characteristics models. Rev Econ Stud 75(2):371–389
Blundell R (2005) How revealing is revealed preference? J Eur Econ Assoc 3(2–3):211–235
Briec W, Kerstens K, Vanden Eeckaut P (2004) Non-convex technologies and cost functions: definitions, duality and nonparametric tests of convexity. J Econ 81(2):155–192
Chambers R, Chung Y, Färe R (1996) Benefit and distance functions. J Econ Theory 70(2):407–419
Chavas J-P, Briec W (2012) On economic efficiency under non-convexity. Econ Theory 50(3):671–701
Chavas J-P, Kim K (2007) Measurement and sources of economies of scope: a primal approach. J Inst Theoret Econ 163(3):411–427
Coggins J, Swinton J (1996) The price of pollution: a dual approach to valuing SO\(_{2}\) allowances. J Environ Econ Manag 30(1):58–72
Dasgupta P, Mähler K-G (2003) The economics of non-convex ecosystems: introduction. Environ Res Econ 26(4):499–525
Duranton G, Turner M (2011) The fundamental law of road congestion: evidence from US cities. Am Econ Rev 101(6):2616–2652
Färe R, Grosskopf S (1983) Measuring congestion in production. J Econ 43(3):257–271
Färe R, Grosskopf S (2000) Decomposing technical efficiency with care. Manag Sci 46(1):167–168
Färe R, Jansson L (1976) Joint inputs and the law of diminishing returns. Zeitschrift für Nationalökonomie 36(3–4):407–416
Färe R, Svensson L (1980) Congestion of production factors. Econometrica 48(7):1745–1753
Ferguson C (1969) The neoclassical theory of production and distribution. Cambridge University Press, Cambridge
First Z, Hackman S, Passy U (1993) Efficiency estimation and duality theory for nonconvex technologies. J Math Econ 22(3):295–307
Fuss M, McFadden D, Mundlak Y (1978) A survey of functional forms in the economic analysis of production. In: Fuss M, McFadden D (eds) Production economics: a dual approach to theory and applications, vol 1. North-Holland, Amsterdam, pp 219–268
Hackman S (2008) Production economics: integrating the microeconomic and engineering perspectives. Springer, Berlin
Henderson D, Russell R (2005) Human capital and convergence: a production-frontier approach. Int Econ Rev 46(4):1167–1205
Jacobsen S (1970) Production correspondences. Econometrica 38(5):754–771
Kuosmanen T (2003) Duality theory of non-convex technologies. J Prod Anal 20(3):273–304
Kuosmanen T (2005) Weak disposability in nonparametric production analysis with undesirable outputs. Am J Agric Econ 87(4):1077–1082
Lau L (1974) Comments. In: Intrilligator M, Kendrick D (eds) Frontiers of quantitative economics, vol II. North-Holland, Amsterdam, pp 176–199
Luenberger D (1992) Benefit function and duality. J Math Econ 21(5):461–481
Luenberger D (1995) Microeconomic theory. McGraw-Hill, Boston
McDonald J (1996) A problem with the decomposition of technical inefficiency into scale and congestion components. Manag Sci 42(3):473–474
McFadden D (1978) Cost, revenue and profit functions. In: Fuss M, McFadden D (eds) Production economics: a dual approach to theory and applications. North-Holland, Amsterdam, pp 3–109
Murty S, Russell RR, Levkoff SB (2012) On modeling pollution-generating technologies. J Environ Econ Manag 64(1):117–135
Paris Q (2008) Law of the minimum. In: Chesworth W (ed) Encyclopedia of soil science. Springer, New York, pp 431–437
Scarf H (1986) Testing for optimality in the absence of convexity. In: Heller W, Starr R, Starrett S (eds) Social choice and public decision making: essays in honor of Kenneth J. Arrow, vol I. Cambridge University Press, Cambridge, pp 117–134
Shephard R (1974) Indirect production functions. Verlag Anton Hain, Meisenheim am Glam
Varian H (1984) The nonparametric approach to production analysis. Econometrica 52(3):579–597
Zhengfei G, Oude Lansink A (2003) Input disposability and efficiency in dutch arable farming. J Agric Econ 54(3):467–478
Author information
Authors and Affiliations
Corresponding author
Additional information
We thank two most constructive referees for most helpful comments. The usual disclaimer applies.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Briec, W., Kerstens, K. & Van de Woestyne, I. Congestion in production correspondences. J Econ 119, 65–90 (2016). https://doi.org/10.1007/s00712-016-0484-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00712-016-0484-6