3.2.1 Damage control model
The C-D production function is the most common form in general factor input production efficiency studies:
\(Y=\alpha \left\{ {\sum\nolimits_{{i=1}}^{n} {\left[ {{{\left( {{X_i}} \right)}^{{\beta _i}}}} \right]} } \right\}{\left( {{X_P}} \right)^\delta }\) (4)
Y is the pig production, XP is the amount of veterinary drug input, Xi is the i-th production factor input other than veterinary drugs that can affect pig production, and \({\alpha },{ {\beta }}_{\text{i}}, {\delta }\) are coefficients to be estimated.
To facilitate the identification of measurements, in this paper we set r = 1 in Eq. (5). Drawing on the expression for the damage control function proposed by Lichtenberg and Zilberman (1986) in an existing study, we obtain:
\(Y=\alpha \left\{ {\sum\nolimits_{{i=1}}^{n} {\left[ {{{\left( {{X_i}} \right)}^{{\beta _i}}}} \right]} } \right\}G{\left( {{X_P}} \right)^\gamma }\) (5)
The above equation can be simplified as:
$$Y=F\left[ {X,G\left( {{X_P}} \right)} \right]$$
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Where F(\(\bullet\)) denotes the C-D production function and G(\(\bullet )\) is the damage control distribution function. G(XP) is defined as a decimation function. We selected the loss control function in the form of exponential distribution based on the principles of simplicity of calculation, ease of understanding and better fit to the data.
Considering that, in the actual pig breeding process, different farmers use a wide range of veterinary drugs with different prices, both injectable and oral, there are practical difficulties in counting the doses and prices of various types of veterinary drugs as a single variable in the research process. Therefore, we constructed the marginal productivity of veterinary drugs by analyzing the relationship between input costs and output benefits. Additionally, in order to compare the differences between the standard C-D production function and the loss control model, we established logistic regression equations for Equations (4) and (5) of the previous section:
\(\ln \left( {{Y_n}} \right)=\alpha +\sum {{\beta _i}\ln \left( {{X_{in}}} \right)+\sum {{\theta _j}} } {M_{jn}}+\delta \ln \left( {{X_P}} \right)+{V_n}\) (7)
\(\ln \left( {{Y_n}} \right)=\alpha +\sum {{\beta _i}\ln \left( {{X_{in}}} \right)+\sum {{\theta _j}} } {M_{jn}}+\ln \left[ {G\left( {{X_P}} \right)} \right]+{V_n}\) (8)
where Yn is the pig farming income of the nth pig farming household; \({\alpha },{\delta },{{\beta }}_{\text{i}},{{\theta }}_{\text{i}}\)are parameters to be estimated; \({\text{V}}_{\text{n}}\) is the random error term; \({\text{X}}_{\text{i}\text{n}}\) is the cost of inputs except veterinary drugs; and M is the control variable. For the standard C-D production function, we adopted a simple ordinary least squares (OLS) regression for estimation. Since the damage control model is nonlinear, we adopted the nonlinear least-squares (NLS) method for estimation.
By taking partial derivatives of Xp for each side of Equations (7) and (8), the marginal productivity of veterinary medicine (MVP) is defined as the farm income generated per unit of additional veterinary medicine. When MVP = 1, the veterinary drug input has reached the economic optimum. The equations are, respectively:
$$MVP=\frac{{\partial Y}}{{\partial {X_P}}}=\frac{Y}{{G\left( {{X_P}} \right)}} \times \frac{{\partial G\left( {{X_P}} \right)}}{{\partial {X_P}}}$$
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\(MVP=\frac{{\partial Y}}{{\partial {X_P}}}=\frac{Y}{{G\left( {{X_P}} \right)}} \times \frac{{\partial G\left( {{X_P}} \right)}}{{\partial {X_P}}}\) (10)
3.2.2 logit model
we constructed a logit model for regression analysis of the influencing factors of veterinary drug over-administration. The specific form is:
\(\ln \frac{{P\left( {{Y_n}=1} \right)}}{{1 - P\left( {{Y_n}=1} \right)}}=\alpha +\sum {{\beta _i}{X_{in}}+{\varepsilon _n}}\) (11)
In Eq. (11), \({\text{Y}}_{\text{n}}\) represents whether the nth pig farmer over-administered veterinary drugs. \({\text{X}}_{\text{i}}\) is the independent variable affecting the over-application of veterinary drugs by pig farmers in different farming regions. In this paper, we drew on the existing research results and used 13 factors in three areas that affect the administration of veterinary drugs by pig farmers as independent variables in the model (as shown in Table 8); that is, individual characteristic factors of pig farmers, including sex (Atreya 2007), age (Si et al. 2022) and level of education (Chen et al. 2016). We also considered pig farmers' household characteristics, including the main business of the household (Ding et al. 2022), the number of household laborers (Min Li et al. 2021), the scale of household farming (Hu and Yu 2022), whether they participate in farmers' cooperatives (Law et al. 2022) and whether they received training in farming (Speksnijder et al. 2015). Government policy factors were considered, including the number of times the relevant government departments supervised the application of veterinary drugs by pig farmers (Milani 2017), the number of fines imposed by the government on farmers for the illegal application of veterinary drugs (Zhao and Hu 2021), the compulsory immunization subsidies granted by the government (Shang et al. 2018) and the number of government-organized campaigns for the reduction in veterinary drug application (Goodhue et al. 2010).