Following the Aigner et al.(1972) and Meeusen and Van dan Broeck (1977) method of estimating a stochastic frontier production function in which the disturbance term (e) is composed of two parts, a systematic term (V) and one-sided (U) component, a Cobb-Douglas production function of the following form was specified.
Q = F(Xa; P) + ei…(1)
Where Q is the quantity of agricultural output, Xa is vector of input quantities, and p is a vector of parameters and ei is defined as
ei = Vi + Ui i = 1, 2,……, n farms…(2)
On the assumption that Ui and Vi are dependent, the parameters of the production frontier (1) were estimated using maximum likelihood method by an econometric software called FRONTIER 4.1. The farm-specific technical efficiency (TEi) of the ith farmer was estimated by using the expectation of Ui conditional on the random variable ei as shown by Battesse and Coeli (1988). That is,
TEi = exp (-Ui )—(3)
So that 0 < T Si < 1
If the functional form of the production frontier is self-dual, for example Cobb-Douglas, then the corresponding cost frontier can be derived analytically and written in general forms as:
C = h(P, Q: у) + ei—(4)
Where C is the minimum cost associated with the production of Q, P is a vector of input prices and y is a vector of parameters.
Allocative efficiency farm i (AE[) is given by
AEt = exp (-Ui)—(5)
So that 0 < AEi < 1
In this cost function, the non-negative random variable Ui which are assumed to account for the cost of inefficiency defines how the farm operates above the cost frontier. If allocative efficiency is assumed (Coelli 1996), the non-negative random variable Ui is closely related to the cost of technical inefficiency.
Following Farrell (1957), equations (3) & (5) can be combined to obtain the economic efficiency (EE) index.
EE = (AE) x (TE)—(6)
To empirically measure efficiency, the first step is to estimate stochastic production frontier and then use the approach introduced by Jondrow et al. (1982) to separate the deviations from the frontier into a random and an efficiency component. To show how this separation is accomplished, consider the stochastic production frontier.
Q = f(Xa; P) + E—(7)
Where E = V + U —(8)
Equation (8) is composed error term (Aigner et al.1977; Meeusen and Van den Broeck 1977). The two components V and U are assumed to be independent of each other, where V is the two-sided, normally distributed random error (V–N (O, a2v), and ^ is the one-sided efficiency component with a half-normal distribution (U— N(O, а2Д The maximum likelihood estimation of equation (7) yields estimators for p and X, where p = vectors of parameters, X = а^ /av and а2 = а2^ + av2.
Jondrow et al. (1982) have shown that the assumption made on the statistical distribution of V and U, mentioned above, makes it possible to calculate the conditional mean of ^i given Si as
E(^i/ £t) = q*rf*(St X /а – Si X!
[1-F* (Si X/а) а —(9)
where F* and f* are, respectively, the standard normal density and distribution functions evaluated at Si X/а and а2 = ац2 ау2/ а2
Therefore, equation (7) and (9) provide estimates for u and v after replacing Si, а, and X by their estimates. If v is now subtracted from both sides of (7), we obtain.
Q* = f (Xi, P) – u = Q – V…(10)
Where Q* is the firm’s observed output adjusted for the statistical noise captured by V.
The use of single – equation model depicted in equation (11) is justified by assuming that farmers maximize expected profits, as is commonly done in studies of this type (Zellner et al. 1966; Kopp and Smith 1980; Caves and Barton, 1990).
For this study, the specific model estimated is
Ln Q = po + Pj LnL + p2 LnF + p3 LnM + p4LnE + p5LnA + Vi – Ui—(11)
Q = the value of production of ith farmer measured in Naira /ha L = Labour used in Mondays/ha (family and Hired)
F = The quantity of fertilizer used, in kilograms/ha
M = the value of material input used in Naira/ha (maize seed, yam Sett and cassava cuttings)
E = the value of implements in naira/ha
A = the quantity of agrochemicals (herbicides and pesticides) used in litres/ha В = Parameters to be estimated
V = is the two sided, normally distributed random error
U = is the one-sided efficiency component with a half-normal distribution.
To examine the possible relationship between efficiency and socio-economic characteristics, multiple regression analysis was used to investigate the association between efficiency indexes and Four Socio-economic characteristics. The level of efficiency, the dependent variable, lies between 0 and 1. The model is specified as:
E 1 = a + bj edu b2 exp + b3 ext + b4 Soc + e
E1 = are the levels of efficiency indexes, economic efficiency (EE), technical efficiency (TE) and allocative efficiency (AE)
Edu = years of schooling completed by the respondent Exp = years the farmer has been in cassava production Ext = number of contact with extension agent.
Soc = equal 1 for membership of social organization and zero otherwise.