Journal of Productivity Analysis

, Volume 38, Issue 1, pp 45–52 | Cite as

Experimental designs for estimating plateau-type production functions and economically optimal input levels

  • B. Wade Brorsen
  • Francisca G.-C. Richter


Estimation of nitrogen response functions has a long history and yet there is still considerable uncertainty about how much nitrogen to apply to agricultural crops. Nitrogen recommendations are usually based on estimation of agronomic production functions that typically use data from designed experiments. Nitrogen experiments, for example, often use equally spaced levels of nitrogen. Past agronomic research is mostly supportive of plateau-type functional forms. The question addressed is if one is willing to accept a specific plateau-type functional form as the true model, what experimental design is the best to use for estimating the production function? The objective is to minimize the variance of the estimated expected profit maximizing level of input. Of particular interest is how well does the commonly used equally-spaced design perform in comparison to the optimal design. Mixed effects models for winter wheat (Triticum aestivium L.) yield are estimated for both Mitscherlich and linear plateau functions. With three design points, one should be high enough to be on the plateau and one should be at zero. The choice of the middle design point makes little difference over a wide range of values. The optimal middle design point is lower for the Mitscherlich functional form than it is for the plateau function. Equally spaced designs with more design points have a similar precision and thus the loss from using a nonoptimal experimental design is small.


c-Optimality Experimental design Linear plateau Mitscherlich Production functions Random effects 

JEL Classification

C93 D2 Q19 



The research was partially supported by the Oklahoma Agricultural Experiment Station. The views stated herein are those of the authors and are not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal Reserve System.


  1. Ackello-Ogutu C, Paris Q, Williams WA (1985) Testing a von Liebig crop response function against polynomial specifications. Am J Agr Econ 67:873–880CrossRefGoogle Scholar
  2. Atkinson A (1996) The usefulness of optimum experimental designs. J R Stat Soc Series B 58:59–76Google Scholar
  3. Atkinson AC, Hines LM (1996) Design for nonlinear and generalized linear models. In: Ghosh R, Rao RE (eds) Handbook of statistics 13. Elsevier, AmsterdamGoogle Scholar
  4. Bäckman ST, Vermeulen S, Taavitsainen VM (1997) Long-term fertilizer field trials: comparison of three mathematical response models. Agri Food Sci Finl 6:151–160Google Scholar
  5. Baule B (1937) Mitscherlich’s Gesets der physiologischen Beziechungen. Landw Jahrb 51:363–385Google Scholar
  6. Berck P, Helfand G (1990) Reconciling the von Liebig and differentiable crop production functions. Am J Agri Econ 72:985–996CrossRefGoogle Scholar
  7. Bornstedt GW, Goldberger AS (1969) On the exact covariance of products of random variables. J Am Stat Assoc 64:1439–1442Google Scholar
  8. Cate RB Jr, Nelson LA (1971) A simple statistical procedure for partitioning soil test correlation data into two classes. Soil Sci Soc Am Proc 35:858–860CrossRefGoogle Scholar
  9. Cerrato MD, Blackmer AM (1990) Comparison of models for describing corn yield response to nitrogen fertilizer. Agron J 82:138–143CrossRefGoogle Scholar
  10. Department of Plant and Soil Sciences (2009) Experiment 502: wheat grain yield response to nitrogen, phosphorus and potassium fertilization, Lahoma OK. Oklahoma State University. Available at
  11. Dette H, Pepelyshev A, Wong WK (2010) Optimal experimental designs for detecting hormesis. Working paper, Department of Biostatistics, University of California at Los Angeles, Available at
  12. Fisher RA (1935) Design of experiments. Oliver and Boyd, EdinburghGoogle Scholar
  13. Frank MD, Beattie BR, Embleton ME (1990) A comparison of alternative crop response models. Am J Agri Econ 54:1–10Google Scholar
  14. Genz A (2004) Numerical computation of rectangular bivariate and trivariate normal and t probabilities. Stat Comput 14:251–260CrossRefGoogle Scholar
  15. Grimm SS, Paris Q, Williams WA (1987) A von Liebig model for water and nitrogen crop response. Western J Agri Econ 12:182–192Google Scholar
  16. Harri A, Erdem C, Coble KH, Knight TO (2009) Crop yield distributions: a reconciliation of previous research and statistical tests for normality. Rev Agri Econ 31:163–182CrossRefGoogle Scholar
  17. Heady EO, Dillon JL (1961) Agricultural production functions. Iowa State University Press, AmesGoogle Scholar
  18. Heady EO, Pesek J (1954) A fertilizer production surface with specification of economic optima for corn grown on calcareous Ida silt loam. J Farm Econ 36:466–482CrossRefGoogle Scholar
  19. Hennessy DA (2009) Crop yield skewness under the law of the minimum technology. Am J Agri Econ 91:197–208CrossRefGoogle Scholar
  20. Hossain I, Epplin FM, Horn GW, Krenzer EG Jr (2004) Wheat production and management practices used by Oklahoma grain and livestock producers. Oklahoma Agricultural Experiment Station Bulletin B-818.Google Scholar
  21. Kaitibie S, Epplin FM, Brorsen BW, Horn GW, Krenzer EG Jr, Paisley SI (2003) Optimal stocking density for dual-purpose winter wheat production. J Agri Appl Econ 35:29–38Google Scholar
  22. Lanzer EA, Paris Q (1981) A new analytical framework for the fertilization problem. Am J Agri Econ 63:93–103CrossRefGoogle Scholar
  23. Liebig J (1840) Organic chemistry in its applications to agriculture and physiology. Bradbury and Evans, LondonGoogle Scholar
  24. Llewelyn RV, Featherstone AM (1997) A comparison of crop production functions using simulated data for irrigated corn in western Kansas. Agri Syst 54:521–538CrossRefGoogle Scholar
  25. Maddala GS, Nelson FD (1974) Maximum likelihood methods of markets in disequilibrium. Econometrics 42:1013–1030CrossRefGoogle Scholar
  26. Makowski D, Lavielle M (2006) Using SAEM to estimate parameters of response to applied fertilizer. J Agri Biol Environ Stat 11:45–60CrossRefGoogle Scholar
  27. Makowski D, Wallach D (2002) It pays to base parameter estimation on a realistic description of model errors. Agronomie 22:179–189CrossRefGoogle Scholar
  28. Mitscherlich EA (1909) Das gesetz des minimums und das gesetz des abnehmenden bodenertrages. Landw Jahrb 38:537–552Google Scholar
  29. Monod H, Makowski D, Sahmoudi M, Wallach D (2002) Optimal experimental designs for estimating models parameters, applied to yield response to nitrogen models. Agronomie 22:229–238CrossRefGoogle Scholar
  30. Paris Q (1992) The von Liebig hypothesis. Am J Agri Econ 74:1019–1028CrossRefGoogle Scholar
  31. Perrin RK (1976) The value of information and the value of theoretical models in crop response research. Am J Agri Econ 58:54–61CrossRefGoogle Scholar
  32. Raun W, Solie J, Johnson G, Stone M, Mullen R, Freeman K, Thomason W, Lukina E (2002) Improving nitrogen use efficiency in cereal grain production with optical sensing and variable rate application. Agron J 94:815–820CrossRefGoogle Scholar
  33. Redman JC, Allen SQ (1954) Some interrelationships of economic and agronomic concepts. J Farm Econ 36:453–465CrossRefGoogle Scholar
  34. Retout S, Mentré F, Bruno R (2002) Fisher information matrix for non-linear mixed-effects models: evaluation and application for optimal design of enoxaparin population pharmacokinetics. Stat Med 21:2623–2639CrossRefGoogle Scholar
  35. Russell EJ (1926) Plant nutrition and crop production. University of California Press, BerkeleyGoogle Scholar
  36. Snedecor GW (1934) Analysis of variance and covariance. Collegiate Press Inc., AmesCrossRefGoogle Scholar
  37. Spillman WJ (1923) Applications of the law of diminishing returns to some fertilizer and feed data. J Farm Econ 5:36–52CrossRefGoogle Scholar
  38. Sumelius J (1993) A response analysis of wheat and barley to nitrogen in Finland. Agri Sci Finl 2:465–479Google Scholar
  39. Tembo G, Brorsen BW, Epplin FM (2003) Linear response stochastic plateau functions Selected paper Southern Agricultural Economics Association annual meeting, Available at
  40. Tembo G, Brorsen BW, Epplin FM, Tostāo E (2008) Crop input response functions with stochastic plateaus. Am J Agri Econ 90:424–434CrossRefGoogle Scholar
  41. Upton M, Dalton G (1976) Linear production response. J Agri Econ 27:253–256CrossRefGoogle Scholar
  42. Wolfinger RD (1993) Laplace’s approximation for nonlinear mixed models. Biometrika 80:791–799CrossRefGoogle Scholar
  43. Wolfinger RD (1999) Fitting nonlinear mixed models with the new NLMIXED procedure SUGI Proceedings. SAS Institute, CaryGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Agricultural EconomicsOklahoma State UniversityStillwaterUSA
  2. 2.CENTRUM Católica, Graduate School of BusinessPontificia Universidad Católica del PerúLimaPeru
  3. 3.Department of Community DevelopmentFederal Reserve Bank of ClevelandClevelandUSA

Personalised recommendations