4.3.1 Building on the Stochastic Profit Frontier Literature
This leads us to the principal methodological approach, which is to estimate a stochastic profit frontier in the usual way, and thus derive a farmspecific measure of relative inefficiency in securing profit per unit of output. Then, following the lead of Jondrow et al. (1982), Ali and Flynn (1989), and Battese and Coelli (1995), the relative profit inefficiency of each farm is explained in terms of the transaction cost barriers and farmspecific policy distortions faced by that farm. The approach thus explains each farm's unit profit performance in terms of technical and allocative efficiency, in a first stage, and then in a second stage explains differences in efficiency in terms of farmspecific differences in transactions costs and policy distortions.
First stage: The stochastic profit frontier. The stochastic profit frontier used in this study to estimate profit efficiency is defined as
Y_{i} = ¦(X_{i}, W_{i}, P_{i}, b) exp(v_{i}  u_{i}) 
(1) 
where Y_{i} = profit per farm i normalized by quantity of output Q_{i} (i.e. unit profit) defined as:
Y_{i} = ((P_{i}Q_{i}  C_{i}Q_{i})/Q_{i}) 
(2) 
where:
P_{i}Q_{i} is total revenue from livestock activity per farm i in question (manure sales included);
C_{i}Q_{i} is total variable costs such as costs on feeds, fodder, day old chicks (DOC) or weanlings, hired labor, electricity, medicines, vaccines, cost, water, depreciation, etc., of securing revenue, excluding family labor per farm i; and
Q_{i} is the quantity of output per farm i;
X_{i} = vector of fixed factors used to obtain Y_{i} (e.g., stock of family labor, land, buildings and equipment, and fixed capital stock, to control for differences in farm resources) normalized by quantity of output Q_{i};
W_{i} = vector of farmspecific input prices;
P_{i} = weightedaverage output price (weights are the farmspecific transaction quantities);
b = vector of unknown parameters to be estimated; and v_{i}, u_{i} are random error terms.
The error term (v) is distributed independently and identically as a twosided normal random variable around the frontier, to account for measurement error on both sides of the frontier, and inefficiency (u) is distributed independently as a onesided (downwards) random variable relative to the frontier, to allow for the fact that farms in fact fall below the ideal efficiency. Average efficiency can easily be estimated by OLS regression of unit profit per farm against farmspecific input and output prices and farmspecific fixed factors of production. However, the frontier, showing ideal unit profits for any given level of farm resources and prevailing price level, can only be estimated if specific assumptions are made about the distributions of u and v across farms, and then only using a nonlinear estimation technique, such as Maximum Likelihood Estimation (MLE). The critical assumption is the distribution of the u (inefficiency) term. We adopt the approach of Battese and Coelli (1995), which allows for systematic differences across farms in the distribution of u, such that we do not assume away what we wish to investigate: viz. that transaction cost factors and policy distortions that are different for different farms help determine their relative profit inefficiency.
Second stage: Technical inefficiency determinant model. Battese and Coelli (1995) base their approach on the assumption that the expected value of the farmspecific inefficiency effect for farm i can modeled as a function of farmspecific characteristics (which of course vary across farms) and fixed coefficients, which do not. In other words, u_{i} ~ N (m_{i}, s^{2}_{u}), where mi = z_{ik}d_{k} is the mean of a truncatednormal distribution of u_{i}. The z_{ik} are k explanatory variables observed for farm i, associated with technical inefficiency effects (u_{i}), and d is a vector of unknown coefficients to be estimated simultaneously with equation (1). Thus, the technical inefficiency effect, u_{i} in equation (1), can then be specified as u_{i} = z_{ik}d_{k }+ e_{i}, where e_{i} is the inefficiency error term, defined by the truncation of the normal distribution with mean equal to zero and variance s^{2}. The truncation of e_{i} occurs at e_{i} ³  z_{ik}d_{k} (Battese and Coelli, 1995).^{[21]}
4.3.2 Specification of an Estimation Model
A translog profit frontier is used because of the flexibility it allows in estimating parameters where it is not desirable to build in through model specification rigid assumptions about substitution relationships among inputs and factors. The full form of the model is:
(3) 

(dummies) (input prices) (fixed factors) 

(pricefactor interactions) (factor interactions)
+ v_{i}  u_{i}
(random error) (random technical inefficiency effect)
where Y_{i} is the normalized profit of the ith farm defined in equation (2); W_{ij} is the price of input j (j= hired labor, capital, feeds, DOC/weanlings, medicines, electricity, etc., used by the ith farm); X_{ik} is the fixed factor k used by the ith farm (k =is the normalized value of breeding stock, normalized value of buildings and equipment, normalized total farm labor in hours, etc.); and D_{il} is a vector of dummy variables for farm i.
Normalizing by output quantity builds in an assumption of constant returns to scale, so to allow for the fact that larger producers may in fact be using highergrade technology than others, we need to control for nonHicks neutral technical change on the RHS of the first stage of our stochastic profit functions. To deal with this problem, we allow for the parameterization of Hicks nonneutral technological change by including the feed conversion ratio (FCR), which is probably closely correlated with both technology and managerial ability. For poultry (broiler and layers) and pigs, we calculate the weighted average FCR per farm as: total feed used (in kg) divided by total output (in kg of liveweight). For dairy, we use total liters of milk per farm and divide by total number of milk animals for that farm. The resulting weightedaverage milk output per animal is a good proxy variable for technological progress.
For hog production, we need to further control for the fact that some farmers do fullcycle operations (farrowtofinish), while others do weantofinish, while still other farms may do both. The total revenue and total cost figures used to compute total revenue before normalization allow for mixing different kinds of sales and costs, as do the average farmspecific output and input prices used on the right hand side (RHS) of the SPF. In order to pool fullcycle and weantofinish producers, we use a binary dummy variable for type of producer on the RHS of the SPF, to allow for fixed differences in mean unit profits among activities on the same farm. The v_{i}, u_{i} are as previously defined. The a_{kj}, j_{kk}, b_{jk} are coefficients to be estimated by MLE using Frontier 4.1 software (Coelli, 1992).
Translog and CobbDouglass profit frontiers share the use of logarithms in the dependent variables, and thus do not handle cases of negative or zero unit profits. Yet it is not unreasonable to suppose that some farms in some years lose money. There is in fact no perfect fix for this problem, and we employ a lesseroftheevils approach that is adequate for present purposes. A constant scalar is added to the unit profit data in each sample such that unit profit of every farm is positive. As long as the cases of negative average farm profits are few (less than 5 percent, say), and they are proportionately not very negative relative to average farm unit profit (so that the scalar is small relative to the mean), the resulting bias from a nonlinear transformation of the data is judged to be of minor importance compared to the bias that would arise from using a less appropriate functional form or arbitrarily dropping the least efficient sample members.
The technical inefficiency effects (u_{i} s) generated in equation (3) are estimated within the MLE model specified above as:
u_{i} = d_{o} + d_{1}Z_{1} + d_{2}Z_{2} ... +e_{i} 
(4) 
where Z_{i} is the ith farm characteristic determining relative inefficiency and e_{i} is distributed as above.
The RHS variables of equation (4), that is the Z, cover all the farm characteristics that proxy different levels of transaction costs faced by each farm, such as access to credit for capital/feeds, existence of policy distortions, subsidized veterinary medicines, access to cheaper sources of feeds, subsidies and taxes (for differences in policy distortions), access to potential and other source of income, access to markets for output, access to information, distance to nearest city or residential area, and other exogenous variables.
Three remaining methodological problems concern the measurement of the farmspecific data. First, some of the explanatory variables that we may wish to include in the second stage may not be observable at all, or in any event very hard to observe. This is especially true of transaction cost and externality variables. Second, some of the explanatory variables that we may wish to include in the second stage may be endogenous in the sense discussed above: the causality goes both ways, introducing simultaneity bias in estimation. This is particularly a problem for environmental externalities, since farmspecific difference here will help to determine relative unit profit efficiency as we define it, but themselves may be a function of the latter in some cases. Third, contract farming may be quite prevalent in some cases, and this raises the issue of which output prices and which input prices to use, as these are accounting concepts rather than actual prices for contract farmers.
^{[21]} The loglikelihood
function of this model is presented in the appendix of Battese and Coelli
(1993). Estimation of the likelihood function also requires the specification of
a relationship among the variance parameters such that (= var(m_{it})/[var(v_{i}) +
var(u_{it})], where the parameter has value between 0 and
1. 