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Land Quality and Agricultural Productivity: A Distance Function Approach

Land Quality and Productivity

Productivity growth (or decline) can be defined as the observed change in outputs over time
relative to the observed change in inputs. For production systems with a single output and a
single input, productivity change can be defined simply as the change over time in the ratio of
output quantity to input quantity. For production systems with multiple inputs and/or multiple

outputs, a method that conveniently aggregates inputs and outputs is necessary. The distance
function is one method that accomplishes such an aggregation. Expanding the definition of the
distance function slightly, we denote the distance from a producer’s input and ou tput levels in
one period, s, relative to the technology defining the frontier in another period, r, by the function

D^r(x_s,ys). The Malmquist productivity index is defined by F are, Grosskopf, and Lovell (1994) as

A value of the MPI greater than one signifies productivity improvement, and a number
less than one denotes productivity decline. This expression can be further decomposed into
terms that distinguish efficiency change and technical change between the two periods:

' D⁺¹⁽ x_t+1^, y_t +1) Y D ⁽ x_t+1^, y_t+1) D ⁽ x_t^, y_t ) Ï

I D (X_t, y_t) ЛD^t+1(x_t+1,y_t+1) D^t+1(x_t, y_t ))

The first term in brackets is the change in efficiency between the two periods. The second term
is the geometric mean of the frontier shifts in each time period, representing the change in
technology.

To evaluate the influence of land quality on agricultural efficiency (hence productivity),
we compute for each observation two groups of distance functions . The first group of distance
functions is computed with respect to all producers. That is, the set I contains all countries in the
data set. These unrestricted distance functions are denoted DU(X,y). The second group of

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