18 June 2010 νi captured random variation in fish production because of factors beyond the control of the farmers, such as variation in weather. The second error term υi, captured technical inefficiencies in fish production. These were assumed to be farmspecific non-negative random variables, i.i.d.~N( , ) u 2 µ σ . A higher value for υi implies an increase in technical inefficiency. If υi was zero the farm was technically efficient. Consequently, technical efficiency (TE) was defined as the ratio of the mean output for the i-th aquaculture farm, given the values of the inputs xi and its technical inefficiency effect υi, to the corresponding mean output if there was no technical inefficiency in production (Battese and Coelli 1988). The definition can be expressed mathematically when yi and xis are in logarithm form as: 2 All estimates of equations 1 and 2 were obtained through maximum likelihood procedures in the computer program FRONTIER 4.1c (Coelli 1996). Measure of input-specific allocative efficiency This study followed a neoclassical production theory approach. Using the farm specific production function with the highest associated iso-profit line, we obtained a measure of input-specific allocative efficiency for the aquaculture farms. The highest iso-profit, however, was determined when marginal value product (MVPx) of the inputs equated marginal factor costs (MFCx). In other words, MVPx was obtained when the slope of the production function (marginal product -MPx) equaled the ratio of the prices of the factor inputs and the output( ) x y MFC P/ 2 (Kalirajan and Obwona 1994). Mathematically: 3 x y x MP .P MFC = 4 where x y x MP .P MVP = x MVPx MFC = 5 For this study, we expressed the derivation of the individual farm specific allocative efficiency for the inputs slightly different from the expressions 3 to 5. This is because of our choice of Cobb-Douglas functional form3 to represent the frontier model (equation 1). However, we derived the individual farm input specific allocative efficiency using the following expression because of the reasons outlined in note 2 as 6 7 Here, βj was the estimated input elasticities (coefficient of the chosen Cobb-Douglas functional form); i ij Y X/ was average product of j-th input; x MFC was price of the factor input j; yP was price of output; ji i ij Y X/ β equivalent to the marginal product ( )x MP of the input. The expression in equation 7 was the measure of the input specific allocative efficiency employed for the study. This was calculated for each variable input per aquaculture farm. The input specific allocative efficiency shows how farmers responded to price signals for output and inputs to allocate their resources (input-mix) in an optimal manner. This might have involved using less of one input or using more of another input to increase their production over time. For an optimal input utilization, marginal value product (MVP) of input xj was expected to equate marginal factor cost (MFC) of the input for an optimum production level to be achieved (i.e. x x MVP =MFC ). However, whenever MVP of an input xj was greater than its MFC (i.e. x x MVP MFC > ) it implied that xj was underutilized in the course of production, thus not used sufficiently. Over utilization of the input was observed when its MVP was less than the MFC (i.e. x x MVP MFC < ). The implications of the last two scenarios signal a non optimum production level. Such characterizations implied continued application of under-utilized inputs as well as decreased application of over utilized inputs to ensure an optimum production level. Model specification For this study, Cobb-Douglas functional form was specified for the study for the reason stated in Note 3. The frontier functional form was, therefore, defined as 8 where, ln represented the natural logarithm; the subscript i-th sample farmer; yi represents the harvested fish (kg) for farmer i ; xj represents pond size, feeds, labour, numbers of fingerlings-stocked and costs of materials; βj represents the input coefficients while iv , and iu are as earlier defined. Emperical Results and Discussion Summary Statistics The summary statistics of variables included in the regressions show that an average farm in Ogun, Ondo, Ekiti and Osun states produced about 23,000, 19,000, 15,000 and 13,000 Kg/yr, respectively. For the inputs, analysis showed that an average farm in Ogun state had about 341 m2 of pond, 4,400 kg of feed used, 1,300 hours of labor, 34,800 of stocked fingerlings and N48,000 costs of materials. An average farm in Ondo state had about 260 m2 of ponds, 3,100kg of feed used, 910 hours of labor; 26,000 fingerlings and N32,000 costs of materials. For an average farm in Ekiti state we observed 210 m2 of ponds, 2,510 Kg of feed used, 968 hours of labor, 14,560 fingerlings stocked and N33,000 costs of materials. Finally, an average farm in Osun state had 194 m2 of ponds, 2,240kg
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