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:

Look at notes in isoquants: implies linear isoquants

Typical alternative:

If focusing on Water, then have

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

Isoquant for bivariate case:

It can be shown: (non-linear):

See problem set key 6b for plot:

Can mix and match single-input sub-production functions

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) and the implicit univariate production function is simply: or and .

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

, where Y is average tuber length, t is thermal time (growing degree days),  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

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

Weibullpdf:

Weibullcdf:

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.

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

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

Cousenseq 3: here YL is % loss, not proportion, so must divide YL by 100 to convert (i.e., 10% to 0.10):

Cousenseq 7: Yield = Pest free yield (1 – ) =

This is the same as Swinton et al. (1994) eq. 1

Swinton et al. eq 2: multi-weed species model of yield loss:

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 xozai per acre, then get kill rate or control rate (% efficacy) against species i:

which is a negative exponential kill or efficacy function with no intercept (no kill at x = 0), then simplify to: 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

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: 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 stdev

and

Hutchison et al. (2010) (SOM): combine all these to get

Harvested yield is then

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?