Modeling Multiple Input Production Functions
Build on Al-Kaisi and Yin (2003)
Yield = f(W,N,D), where W = water (ET), N = N rate, and D = plant density
For estimation, have to specify a model, what model specify?
A “joint” function: y  ymax 1  exp(  wW   N N   D D 
Look at notes in isoquants: implies linear isoquants
Typical alternative: y  ymax f (W ) g ( N )h( D)
If focusing on Water, then have y   ymax g ( N )h( D) f (W )  Ymax f (W )
Can do this for each input. So if focus on a single input, the other inputs determine the
maximum yield or yield potential and the single-input function (i.e., f(W), g(N) and h(D))
determines the proportion of this yield max obtained
Example: multivariate negative exponential
y  ymax i f i ( X i )  ymax i 1  exp(i  i X i ) 
y  ymax 1  exp(W  WW ) 1  exp( N   N N ) 1  exp( D   D D) 
Isoquant for bivariate case: y  ymax 1  exp( N   N N ) 1  exp( D   D D) 


1  
y
ln 1 
   N  (non-linear):
 N   ymax (1  exp( D   D D)) 

See problem set key 6b for plot:
It can be shown: N 
Can mix and match single-input sub-production functions


D
y  ymax  min(W  WW , ymax ) 1  exp( N   N N )  

 1   D D / D 
Generally use this approach, where each single-input sub-production function is based on
empirical literature or biological theory for each sub-field. However, can specify a joint function
if have a biological basis for this function.
Conceptually, for input D, it is the case that the ymax is a function of the other inputs (W and N)


D
and the implicit univariate production function is simply: y  ymax W , N  
 or
 1  D D / D 
y  ymax W , D 1  exp( N   N N )  and y  ymax  N , D  min(W  WW , ymax )  .
This leads to the next way to think of these production functions
Hierarchical Models: Model the underlying production process biologically and then the inputs
affect the parameters of the biological model.
Goeser et al. (2012): crop development during the season as a logistic function
Y (t ) 

, where Y is average tuber length, t is thermal time (growing degree days),
  t 
1  exp 

  
 is maximum,  is inflection point (where reaches ½ maximum), and  is scaling factor: how
fast reaches maximum
Here:  = 100,  = 25,  = 5
Data for Goeser et al.: average tuber length and
Average number of tubers per plant
Problem: what’s the input the farmer chooses?
We used Stems, which is essentially Density
Key: we look at data and model the parameters of the logistic response function as functions of
the Stems density: how the plant responds to GDD depends on the stem density.
Separated data into small groups based on stems per pant, then estimated the alpha, beta and
gamma and plotted the results to see what functions made sense:
Figure 1:
See a “linear” response for 
and no clear response for 
These helped us choose functions to use,
but actually estimated model parameters
simultaneously.
Model estimated in R using non-linear least squares
AverageTuberLength 
 0   stems Stems


 t
1  exp 

  0   stems Stems 
Table 2 reports results, pooling across years
Table 3: each year separately, see often not
different between years.
Figures 2 and 3: plots of the data in 3-D.
Copas et al. (2007): Fitting the tuber size distribution as a function of Stem Density
Fit Weibull model to describe tuber size distribution, then have stems affect the parameters of the
Weibull distribution

Weibull pdf: f ( x)   x 1 exp   x /  


Weibull cdf: F ( x)  1  exp   x /  


/
a
Fit the cdf to estimate parameters a and b as functions of stem density: basically fitting a
negative exponential model that has an “S” curve and no intercept. F ( x)  1  exp   x 
Main idea: Hierarchical Model: model the biological process, then the parameters of the
biological model depend eon other factors, often manager’s choice.
Intermediate Inputs
Many production processes can be better thought of and modeled with intermediate inputs
y  f ( x1 , x2 )  f ( g ( x1 ), x2 )  f ( z, x), where z  g ( x1 )
z = plant N uptake, x1 = applied N
z = seeds planted, x1 = stand
z = stalk tunneling, x1 = ECB per plant
z = NIS, x1 = CRW larvae per plant
Typically linear models for N and seeds:
N applied versus N available: estimated model with field data was E[NAvailable] = N0 + kNApplied
 Babcock et al. (1996): focus on the underlying probability distribution of Navailable based
on N applied, not the expected value
 Assume constant germination rate, say 96%: Stand = 0.96 x SeedRate
Pest models: often model the loss, then create yield model
% loss = f(weed density)
Cousens (1985)
% loss = f(NIS)
Tinsley et al. 2012
Yharvest = Ypestfree(1 – %loss) = Ypestfree(1 – %loss(WeedDensity))
Problem: what’s the weed density or pest damage? Need to model the control process
Id
here YL is % loss, not proportion, so must divide YL by 100 to
1  Id / A
Id
convert (i.e., 10% to 0.10):  
100(1  Id / A)
Cousens eq 3: YL 


Id
Cousens eq 7: Yield = Pest free yield (1 – ) = y  y pf 1 

 100(1  Id / A) 
This is the same as Swinton et al. (1994) eq. 1


 i I i di
Swinton et al. eq 2: multi-weed species model of yield loss: y  y pf 1 

 100(1   I i di / A) 
i


Notice: each weed species has a density di and an interference or index of competitiveness Ii.
Also note that the maximum yield A does not depend on the weed species: but you could make it
depend on other variables or inputs, such as N rate or seeding density, or GDD, etc.
What does weed control do? It reduces the density of weed species differentially, so model
effect of control on di: Assume apply compound at rate x oz ai per acre, then get kill rate or
control rate (% efficacy) against species i: di  di 0 (1  ki ( x))  di 0 (1  [1  exp( i x)])
which is a negative exponential kill or efficacy function with no intercept (no kill at x = 0), then
simplify to: di  di 0 exp(i x) which is essentially an exponential decay function with the decay
rate depending on the amount of herbicide applied and a species-specific efficacy parameter i.
Here is i = -0.1 and d0 = 100
Now substitute this control model into the
production function to get yield as a function
of the herbicide application rate x




 i I i di
i Ii d0i exp(i x)
y  y pf 1 
  y pf 1 

 100(1   I i di / A) 
 100(1   I i d 0i exp( i x) / A) 
i
i




You could parameterize this model with real data by estimating the bi using data from herbicide
efficacy trials and Ii using field data that varied populations of key weed species (see Cousens).
Insect Version
Estimate % loss as a function of feeding damage, and then feeding damage as a function of pest
population density, and then pest population density as function of pest control.
Hurley et al. (2004): yield loss as a function of ECB
larval tunneling:   0.21T 0.58  0.058 where  is
proportion of pest free yield lost and T is ECB larval
tunneling/plant (cm) and  is error
Data from 22 IA counties, Bt and non-Bt plots,
Measured tunneling and yield in side-by-side plots
1997-1999.
Mitchell et al. (2002): tunneling cm given ECB larval
population density
 Data from 9 states, average ecb/plant and tunneling
in 1997 (211 obs).
 Tunneling has lognormal pdf with mean and st dev
T  2.56n  5.65n0.5 and  T  3.40  1.73n
Hutchison et al. (2010) (SOM): combine all these to get
 (n)  0.021
(2.56n  5.65 n )1.16

 2.56n  5.65 n


2
2
  3.40  1.73n  

0.29
Harvested yield is then y  y pf (1   (n))
Could then have n depend on control, such as use of Bt corn, or rate of insecticide use x, so that n
= n0exp(x) or n = 0 if bt and n = n0 if conventional.
Note that this ecb model seems rather ad hoc. I’m in the process of working with some
ecologists and entomologists to develop a model with clearer biological foundation: How does
tunneling affect yield loss? How do ecb larvae tunnel, especially if multiple ecb/plant?