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Land Quality and Agricultural Productivity: A Distance Function Approach

The relative technical efficiency of producers can be characterized by a distance function.

Following Fare, Gosskopf and Lovell (1994), we define an input set X(y) as the set of all inputs x
that can produce a given output level y. The input distance function D(x,y) is a scalar value that
describes the maximum proportional decrease in inputs achievable for a given level of output so
that the input level remains in the set X(y).

The input-oriented distance from an observation to the frontier for a production system
using J inputs and K outputs can be expressed as the optimal solution to the following linear
program:

D ( x, У ) = min θ d                                                                          (1)

subject to the constraints

∑^λi^xj ≤ ^θd^xd^{,         j} = ^1,..., J

i ∈ I

∑ ⅜y. ≤ Ук,      k = U. K                                             (2)

i∈ I

Λ _i ≥ 0

where the set I contains all the observations that are eligible to define the frontier relative to
observation d, and x_dj and y_dk contain the input and output data elements for the observation
being evaluated, respectively. The inputs and outputs for each observation are contained in the
parameters x_ij and y_ik. The optimal solution to the linear program for each observation d includes
the value θ_d, which takes values between zero and one. A value of θ_d equal to one means that the
observation lies on the frontier—it is technically efficient. The observations for which λ_i is
greater than zero determine the part of the frontier with which the country is being evaluated. It
is relative to the performance of these observations that the technical efficiency of observation d
is measured. This mathematical program is referred to as an input-oriented, constant-returns-to-
scale data-envelopment-analysis (DEA) model.

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