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A Bayesian Method for Plateau Type Models

Back in 2017 when I was just starting out as an Assistant Professor, I met my colleague (and now good friend) Ford Ramsey. Ford and I have similar interest in traditional agricultural economics research problems and, like the academics we are, were discussing production functions at the departmental picnic. The first class I ever attended as a Ph.D. student was Dr. Wade Brorsen’s Advanced Production Economics class were we learned about plateau functions in the first few weeks. Talk about a quick and intense introduction to Ph.D. classes – I struggled to keep up and change my mindset to Dr. Brorsen’s teaching methods. But what intrigued me well after that class was the difficulty in estimating the nonlinear relationship of plateau (also known as von Leibig) type models. In fact, one of my favorite questions to ask my undergraduate intermediate micro theory class was, “why can’t we just keep adding fertilizer to a wheat plant and increase output?” Well, at some point, inputs like nitrogen, phosphorus, potassium, and even water are no longer effective in increasing output – they reach a limit of usefulness.

Graphically, we end up with a “kinked” regression that is often modelled using nonlinear equations. So you get something like Figure 1 in our paper:

When estimating these with frequentist methods, the issue is sensitive results and finicky convergence. Just trying to get the model to converge depends on starting values, scaling, optimization procedures and criterion, and a host of other things. I spent weeks (literally) trying to solve the homework Dr. Brorsen would assign on the topics. So, Ford and I spent some time talking about how to get around these issues and came to the conclusion that Bayesian methods would allow for a more robust solution to the problem. To help us solve the Bayesian problem, we approached Klaus Moeltner with our idea and agreed to join in on the project. Klaus’s passion for Bayesian econometrics really helped to develop our method and inspired me to learn and use Bayesian methods in other work (hopefully forthcoming in a paper one of these days).

As with all good ideas, there were already a couple of Bayesian approaches to the problem in the literature. Specifically, the mixture model by Holloway and Paris (2002) and the Hierarchical approach by Ouedraogo and Brorsen (2018). However, we noticed that these other approaches were focused on estimating several parameters as in the frequentist approach. As we show in our paper, you really only need to estimate the threshold value of the input parameter and the other sampled quantities (i.e. plateau level) can be recovered deterministically. Moreover, this approach allows for a robust estimation of multiple inputs, which is a point that has lacked in the literature.

While there is a lot of math to this paper, I will skip let you work through it on your own. For now, I’ll skip to the application. We use some publicly available corn yield data and compare our model to that of Holloway and Paris (2002). When we apply the results to profit functions. What we find is that our proposed model suggests the least amount of nitrogen application and the largest profit for the producer. We also show that incorrect model choice could have large implications for producers. For example, if the mixture model is the true model, but the threshold model estimates are used then the producer loss is 1.8 times that if the reverse was true. So, if the producer is wrong, it is better (though still worse-off) to choose the mixture model. However, we find that our proposed threshold model has a better fit with the actual data.

Overall, this work is a culmination of an idea born of two newly minted assistant professors and a fantastic econometrician. Our goal with this paper was to decrease the difficulty in estimating plateau type models via a Bayesian method. Our proposed threshold model requires less parameters and doesn’t heavily depend on starting values, scaling, etc. for reaching convergence – a lot less frustrated hair pulling in my opinion. An ungated copy of the paper can be found with the below link. Hope you find this new method useful for all of your von Leibig problems!

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