Land Quality and Agricultural Productivity: A Distance Function Approach
distance functions is computed for each country only with respect to those countries for which
the land quality index is less than or equal to that of the given producer. These land quality-
limited distance functions are denoted DL(x,y). The degree to which the two frontiers are similar
is indicated by the ratio of the two efficiency measures:
D DU ( x, У )
(5)
LQ Dl ( x, У )
Since the unrestricted frontier always lies above the limited frontier, an observation will
always be farther from the unrestricted frontier than it is from the limited frontier, and thus less
efficient relative to the unrestricted frontier than it is relative to the limited frontier. Therefore
Du(xy) is always less than or equal to Dl(x,y). Since both values are always greater than zero, it
follows that 0 ≤ alq ≤ 1. A value of alq = 1 implies that the producer is efficient, or equally
inefficient, with respect to both frontiers. In this case land quality does not contribute directly to
inefficiency; all inefficiency is attributable to input use. A value of aLQ that is less than one
implies that a portion of the inefficiency measured by DU can be attributed to poor land quality.
Since alq measures the agreement between the two frontiers, the percentage difference between
the unconstrained and limited measures is given by 100*(1-αLQ).
Substituting aLQ Dl (x,y ) = Du(x,y ) into equation 4 (with Du(x,y ) taking the place of
d (x,y )) allows further decomposition of the MPI into land quality, technical change, and
efficiency change components:
t rt,t + 1 /"Vt+1,t+1 A t T~λt +1 / ∖ ∖Λ 7^Λ t / ∖ τ~∖t / ∖ ʌ1
M ( xt, yt, xt+1, yt+1)
a LQ a LQ dl ( xt +1, yt+1) I dl ( xt+1, yt+1) dl ( xt, yt ) I
(6)
fvt,t ™t+1,t Dt (x V ) J Dt+1 (x V ) Dt+1 (x V ) J
V txLQ uLQ J V dl (xt, yt ) ʌdl (xt+1, yt +1) dl (xt, yt ) J
(Productivity growth) = (Land quality contribution)*(Efficiency change)*(Tech. change)