Hi,

Kindly refer the following link and you will get some idea to your study.

https://ideas.repec.org/r/spr/empeco/v20y1995i2p325-32.html

Several agricultural economics studies have addressed the damage reducing role of pesticides versus the productivity-increasing role of other inputs.

If we assume a Cobb-Douglas production function and use its log linear specification we have:

 ln Q = α+ βln Z + γln X

 where Z denotes the productive inputs and X  pesticide inputs. 

 Lichtenberg and Zilberman (1986) were the first to provide an econometric approach for using an abatement relationship for pesticides. Their suggestion is to capture the pesticide variable in an abatement function and then to enter the abatement function rather than the pesticide variable into the production function. Thus Lichtenberg and Zilberman’s remedy is to replace X with an abatement function, G(X), which shows the proportion of the destructive capacity of the damaging pest, weed or disease that is eliminated by the application of a level of control agent X, i.e. pesticides.  

Different damage abatement functions can be used, for example logistic or Weibull. There is no agreement on the most suitable specification of the damage control function.  Empirical studies show that results can be sensitive to the functional form.

In order to compare the results with those from the traditional approach, the  approach is to first estimate the coventional  production function above using ordinary least squares (OLS), where pesticide use are specified as other inputs such as labour and fertiliser. The results are then compared with those from a Lichtenberg and Zilberman specification which is estimated using non-linear methods.  See for example   http://hdl.handle.net/10355/295. http://ageconsearch.umn.edu/record/31062/files/28030596.pdf

Based on insights provided by production ecology,   Wossink and Rossing (1998) argue that it is not just pesticides that act indirectly on output — this also applies to plant nutrients. Guan et al. (2006) further elaborate on this theme and develop a general framework that distinguishes between growth inputs and facilitating inputs.

Article Integrating Agronomic Principles into Production Function Sp...

Article On increasing returns and discrete choice: Integrating produ...

Article The Econometrics of Damage Control: Why Specification Matters

Thanks Dr Senthilvel and Ada Wossink

I have gone through all the literature that you have shared before.

I am not knowing how to do the analysis esp nonlinear least square procedure ie, damage abatement functions and integrate the two functions

As highlighted above, the traditional production function form used to estimate productivity of pesticide inputs is the Cobb-Douglas function.

For the damage abatement function for the estimation of pesticide productivity alternative specifications exist and there is no agreement. Commonly used are the exponential and the logistic function.

In mathematical terms this leads to the following specifications:

(1) Cobb-Douglas:     lnY = α + βilnZi + γlnXi

(2) Exponential:           lnY = α + βilnZi + γln[1-exp(-λXi)]

(3) Logistic:                  lnY = α+  βilnZi+  γln[1+exp(μ-σXi)]-1

where   Y is the value of output ; Zi are the production inputs such as seed, fertiliser, and labour; Xi are the damage control inputs,i.e. pesticides.

The parameter restriction y = 1 is imposed on eqn (2) -(3).  This restriction means that damage abatement be proportional and  it facilitates estimation (Carrasco-Tauber and Mofftt, 1992). 

https://www.uni-hohenheim.de/fileadmin/einrichtungen/sfb564/events/uplands2002/Full-Pap-S3B-2_Praneetvatakul.pdf