infrastructure ( _e.g._ that the labor and energy required for operating a STS gantry crane is the same across ports in LAC).

On a different note, working hours of ports in the region could also play a role in the identification of model parameters. If the number of weekly hours of port operations varied, it would be necessary to normalize the use of port infrastructure per working hours. However, according to the figures from Containerization International, all terminals in the sample were open to business 24 hours a day, seven days a week. As a result, working hours do not impact the estimations: occasional idle infrastructure is part of the technical inefficiency once the port can be continuously open for business.

Representing a departure from the standard port production function usually employed in the literature, our database allows us to breakdown the type of cranes owned by a port between mobile/quay cranes (with container handling capacity) and STS gantry cranes. Clearly, these two inputs are expected to have different impacts on throughput<sup>5</sup> , since a typical STS gantry crane in LAC can move at least 50% more containers per hours than a typical mobile crane (Kent, 2011). In the model, we identified these two variables as _MQCit_ and _GCit_ , respectively.

A challenge related to the use of these inputs is that a Cobb-Douglas production function fails to capture the effects of variables when their values are zeros. In our model, _MQCit_ and _GCit_ are nonessential inputs, since container terminals might use, alternatively, mobile cranes, STS gantry cranes or even ships’ cranes to move containers. As a result, in a translog model, observations with zero nonessential inputs would drop out of the sample because the log of zero is unidentified. In our sample, a total of 42 ports doesn’t have either mobile/quay cranes or STS gantry cranes in 2009, and 8 of these ports had no cranes whatsoever. In the literature, different solutions have been assessed, such as using quadratic functional forms, resampling techniques, or substituting a small value for the zero observations (Moss, 2000). According to Soloaga (2000), adding a constant to the va <mark>riable for every observation in the sample can be a solution with little impact to the parameter estimation, depending on the degree of essentiality of the variable. Therefore, in</mark> order to maintain the original sample with ports that have zero non-essential inputs, we added on <mark>e unit to</mark> _<mark>MQC</mark> it_ <mark>and</mark> _<mark>GC</mark> it_ <mark>.</mark> Equation 2 shows the translog model after modifications.

ln(Qit) = α+ β1ln(Ait) + β2ln(Bit) + β3ln(MQCit) + β4ln(GCit) + β5Tt + vit + uit (2)

# **a. Use of Ships’ Cranes**

There are two possible ways to offload containers from ships to terminals: using cranes in the terminal or cranes mounted directly on ships. Therefore, the use of ships’ cranes has to be taken into account when estimating port efficiency because they represent a port-exogenous asset that is fundamental for the productivity of terminals with modest infrastructure ( _i.e._ container ports that do not have any crane, or have just a few, but have a relatively high level of throughput). As a result, disregarding the use of these “shared” assets would benefit the technical efficiency of ports that often rely on ships’ cranes to handle containers whenever this input is omitted as an explanatory variable in the estimation.

In order to account for this heterogeneity, we created a dummy variable that takes the value of 1 when ports are likely to make an intense use of ships’ cranes to handle containers. Thus, this

> 5 Further disaggregation of crane information, for example, by equipment age or crane reliability, is not possible due to data limitations, although these characteristics also play a role in the explanation of crane productivity.

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