ABSTRACT
Transformation processes with multiple inputs typically exhibit nonlinearities in their output with respect to input usages. They have been traditionally modeled via production functions in the microeconomics literature (Heathfield and Wibe, 1987). One of the most common production functions is the Cobb-Douglas (C-D) production function. This production function assumes that multiple (n) inputs (also called factors or resources) are needed for output, Q, and they may be substituted to take advantage of the marginal cost differentials. In general, it has the form Q A x i
n i ,= ⎡
⎣
⎤
⎦
∏ α1 where A represents the total factor productivity of the process given the technology level, x(i) denotes the amount of input i used, and αi > 0 is the input elasticity. The total
13.1 Introduction 271 13.2 Model 274
13.2.1 Assumptions 274 13.2.2 Formulations 276
13.3 Numerical Study 279 13.3.1 Overall Assessment 280 13.3.2 ANOVA Assessment 282
Acknowledgment 287 References 287
elasticity parameter r i
=
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
1 α
may be greater than (smaller than)
or equal to 1 depending on whether there is diminishing (increasing) returns to resources, resulting in convex (concave) operational costs. The C-D production function was first introduced to model the labor and capital substitution effects for the US manufacturing industries in the early twentieth century (Cobb and Douglas, 1928). Despite its macroeconomic origins, since then, it has been widely applied to individual transformation processes at the microeconomic level, as well. For example, the C-D production function was employed to model production processes in the steel and oil industries by Shadbegian and Gray (2005) and in agriculture by Hatirli et al. (2006). Logistics activities associated with shipment preparation, transportation/delivery, and cargo handling also use, directly and/or indirectly, multiple resources such as labor, capital, machinery, materials, energy, and information technology. Therefore, it is not surprising that there is a growing literature on the successful applications of the C-D-type production functions to model the operations in the logistics and supply chain management context. Chang’s (1978) work seems to be the earliest to construct a C-D production function to analyze the productivity and capacity expansion options of a seaport. Rekers et al. (1990) estimate a C-D production function for port terminals and specifically model cargo handling service. In a similar vein, Tongzon (1993) and Lightfoot et al. (2012) consider cargo handling processes at container terminals for their production functions. In a recent work, Cheung and Yip (2011) analyze the overall port output via a C-D production function. Studies on technical efficiency in cargo handling and port operations provide additional support for the C-D-type functional relationships, where output is typically measured in volume of traffic (in terms of twenty-foot equivalent unit-TEUs) and inputs may be as diverse as number or net usage time of cranes, types of cranes, number of tug boats, number of workers or gangs, length and surface of the terminals, berth usage, volume carried by land per berth, and energy (e.g., Notteboom et al. 2000, Cullinane 2002, Estache et al. 2002, Cullinane et al. 2002, 2006, Cullinane and Song 2003, 2006, Tongzon and Heng 2005). Comprehensive surveys can be found in
Maria Manuela Gonzalez and Lourdes Trujillo (2009), Trujillo and Diaz (2003), Tovar et al. (2007), and Gonzalez and Trujillo (2009). For land transportation, we may cite the evidence from Williams (1979) and for supply chain management, Ingene and Lusch (1999) and Kogan and Tapiero (2009).
