Agricultural Productivity and technological gap between MENA ...

kwhere Y stands for the output level, x for the vector of inputs’ levels. β is the vector ofparameters for the kth region. vi( k)’s are iid N(0, σvk ( )). uik ( )is assumed a truncation at zeroof the N ( µi( k) , σu( k)) random variable.The metafrontier production function model for units in the industry is expressed by:Y = f x i = N t = T(2)*it(it, β ), 1,2,...k; 1, 2,...where*β denotes the vector of parameters for the metafrontier function such thatx β ≥ x β(3)it* kit(3) states that the metafrontier dominates all the regional frontiers. That is themetatechnology describes the unconstrained best practice according to the state of knowledge.The observed ouput for any unit at any time period can be consequently defined by thestochastic frontier for the kth region (equation (1)) and using the metafrontier function (2),such that:kxitβ*− u eit ( k ) xitβ+ vit ( k )*xitβYit= e e(4)ewhere the first term on the right-hand side of equation (4) is the technical efficiencyrelative to the stochastic frontier for the kth region. The second term on the right-hand side ofequation (4) is the technology gap ratio (TRG) for the ith unit in the kth region at the tth timeperiod.xitβk eTRGit=*(5)xitβeThe third right-hand side of equation (4) is a pseudo stochastic metafrontier since the error*xterms of the K regions vit( k)is added to the deterministic metafrontier e β it. Dividing bothterms by the pseudo stochastic metafrontier, one can retrieve the famous equality presented byRao, Odonnell and Battese, (2003):k* kkTE ( x, y) = TE ( x, y) TGR ( x, y)(6)it it itEquation (6) implies that the technical efficiency ratio of the ith unit at the tth time periodrelative to the metafrontier (which is a common feature to all the K regions) is the product ofthe technical efficiency ratio relative to the regional frontier and the technological Gap Ratiorelative to the metafrontier. In other words, the technical efficiency scores for units that don’tproduce under the same technology can be corrected (to make them comparable) using thedistance between the regional frontier and the leading metafrontier. For example, if TE=0.9then the technical efficiency measure indicates that the observed output is 90 per cent of the4