The procedure is used to test experimental data on com and wheat. An implication of this hypothesis is that crop production functions exhibit square isoquants.
I Perrin compared optimal fertilizer recommendations derived from von Liebig and standard production theory. Paris used nonnested hypothesis tests to test among five alternative production functions. His conclusion was that the von Liebig model with Mitscherlich regimes best interpreted the experimental data. Berck and Helfand showed how a micro-level von Liebig response could give rise to an aggregated smooth production function. Llewelyn and Featherstone used the CERES- maize simulator to produce data to evaluate production functions and fmd evidence for both a Mitscherlich-Baule formulation and a nonlinear von Liebig.
Chambers and Lichtenberg took a nonparametric approach, constructing an outer and inner approximation to the data and then testing for the presence of yield plateaus and nonsubstitutability of inputs. This paper presents a nonparametric estimation of von Liebig response funet! The estimation does not require approximation and is free of assumptions about functional form.
The von Liebig hypothesis is testable against the composite alternate hypothesis of any other production function because any production function consistent with the von Liebig hypothesis has right-angle also called square isoquants. Agronomic experiments for nutrient response use incomplete block designs.
These experimental designs include many observations that differ in only the amount of one nutrient applied. Given the position of the von Liebig expansion path relative to input-level combinations, the experiments allow direct observation of multiple' points on the same isoquant. It is reasonable to assume that such points will differ only by experimental error. There are four sections of this paPer of which this is the frrst. In the next section, the data and the experimental design that led to that data are described.
In the third section, the estimation and test methods are discussed. The fourth section includes the results of estimation and testing and is the conclusion. A Nonparametric Test: The Theory The von Liebig hypothesis, when applied to experimental data from an incomplete block design, can be tested with a set of restrictions on a regression of grain yield on dummy variables.
These tests depend upon the details of the construction of incomplete block design experiments, which are described in the next subsection. They also depend upon the nonsubstitution or square isoquant property of the von Liebig described next. Putting together, in the fourth subsection, the nature of the experiments and the square isoquant hypothesis, it is possible to characterize all possible orderings of predicted yields that are consistent with theory.
These orderings are finite, and the succeeding section shows how to place an upper bound on the number of them. Finally, the testing methodology is described. Experimental Design The agronomic experiments in question-and many others as well-were conducted using an incomplete block design.
A crop was grown on a number of experimental plots using different combinations of inputs. In the designs considered here, there were two inputs-water and nitrogen-that were applied in up to five different levels for a total of 25 different possible combinations of treatments.
More generally, there would be two 2 inputs applied at n different levels for up to a total of n different possible combinations.
When all points in the grid represent input levels used in the experiment, the experimental design is said to be complete. The agronomic experiments that provide the data for the estimations described below utilize an incomplete block design with only 13 different treatments. That is, 13 combinations of two nutrients were applied to experimental plots and the yields from those plots were measured.
Figure 1 illustrates a grid for the case of an incomplete block design where the 13 solid squares represent the included data points. Quantities of the two inputs-water and nitrogen-are on the axes. A mean output level-yield of wheat for instance-also corresponds to each of these points but is not shown. The data used for this paper are from incomplete designs, so the case of complete designs will be treated only parenthetically.
Application to Wheat and Corn Production Experiments The data for this paper come from the appendix of Hexem and Heady and reflect experiments in several Western states to measure crop response to water and nitrogen over a variety of soil and climate conditions. Among the crops they chose, we have limited ourselves to those that are determinate in their flowering-wheat and com-since it is these crops that are believed to follow von Liebig's law of the minimum.
An experiment recorded either one or two years of production data on a particular type of crop and place e. Generally, 44 plots were part of each experiment allowing for some input combinations to be replicated twice and others four times. Null hypothesis The null hypothesis in this study is that yields are detennined by input levels and a normal error teIDl. Estimation under the null hypothesis is achieved by simply regressing the observed yields on 13 dummy variables--one for each of the 13 different input combinations.
The error term from this regression is the experimental error and should be normally distributed. The von Liebig hypothesis will be shown in the next section to be a restriction on the values that the dummy variables may take. The von Liebig Hypothesis Von Liebig hypothesized that agricultural production would be determined by a limiting nutrient.
This hypothesis l treated in historical detail by Paris, implies that the agricultural production function should have isoquants with square comers located on the expansion path.
With inputs Xl and Xl' plateau level P, and output Y, the formal representation of the generalized von Liebig production function Paris and Knapp is given by 1 where u is an error term and f is an increasing continuous function. After adding sufficient quantities of both inputs, the plateau is eventually reached which could be interpreted as a right-angle isoquant in a higher dimension.
Once the plateau is reached, increasing either input leaves output unchanged. The expansion path is the set of cost-minimizing input combinations.
Since all isoquants have "comers," any isoquant will intersect an isocost line with strictly positive prices at the comer of the isoquant. The expansion path consists of the set of all such comers. By the assumption that production is nondecreasing in inputs, the expansion path must not slope downward.
The lighter vertical and horizontal lines are isoquants or parts of isoquants. A little farther up the expansion path is the horizontal leg of the isoquant intersecting the expansion path at 2, 3. Three further parts of isoquants are shown higher up the expansion path.
The path shown is a stylized representation of the collection of all expansion paths that pass between the data points in the order depicted. The limited number of input-level combinations offers no information about the exact position of an expansion path between data points.
An upward-sloping expansion path and its associated isoquants contain all of the restrictions imposed by the von Liebig hypothesis. In terms of the data points, the predicted output level at a point is the same as at another point if they are on the same isoquant and higher if on a higher isoquant.
Thus, drawing an expansion path is equivalent to ordering the predicted grain yields of the experimental treatments. For instance, in figure 1, the input combinations 3, 5 and 5, 5 have equal output and output greater than the input combination 4, 4. To each possible expansion path, there is an ordering of the predicted outputs. The least restrictive statistical model that preserves the ordering is a restricted regression of yield on dummy variables. If a system satisfies the law of the minimum then adaptation will equalize the load of different factors because the adaptation resource will be allocated for compensation of limitation.
Indeed, in well-adapted systems the limiting factor should be compensated as far as possible. This observation follows the concept of resource competition and fitness maximization. Inversely, if artificial systems demonstrate significant violation of the law of the minimum, then we can expect that under natural conditions adaptation will compensate this violation. In a limited system life will adjust as an evolution of what came before.
Biotechnological innovations are thus able to extend the limits for growth in species by an increment until a new limiting factor is established, which can then be challenged through technological innovation.
Theoretically there is no limit to the number of possible increments towards an unknown productivity limit. It may be worth adding that biotechnology itself is totally dependent on external sources of natural capital. At the age of 13, Liebig lived through the year without a summer , when the majority of food crops in the Northern Hemisphere were destroyed by a volcanic winter.
Due in part to Liebig's innovations in fertilizers and agriculture, the famine became known as "the last great subsistence crisis in the Western world". He worked with his father for the next two years,  :7—8 then attended the University of Bonn , studying under Karl Wilhelm Gottlob Kastner , his father's business associate. When Kastner moved to the University of Erlangen , Liebig followed him. The circumstances are clouded by possible scandal.
Liebig's doctorate from Erlangen was conferred on 23 June , a considerable time after he left, as a result of Kastner's intervention on his behalf. Kastner pleaded that the requirement of a dissertation be waived, and the degree granted in absentia.
On 26 May , at the age of 21 and with Humboldt's recommendation, Liebig became a professor extraordinarius at the University of Giessen. He received a small stipend, without laboratory funding or access to facilities. Vogt was happy to support a reorganization in which pharmacy was taught by Liebig and became the responsibility of the faculty of arts, rather than the faculty of medicine. Zimmermann found himself competing unsuccessfully with Liebig for students and their lecture fees. He refused to allow Liebig to use existing space and equipment, and finally committed suicide on 19 July The deaths of Zimmermann and a Professor Blumhof who taught technology and mining opened the way for Liebig to apply for a full professorship.
Liebig was appointed to the Ordentlicher chair in chemistry on 7 December , receiving a considerably increased salary and a laboratory allowance. They had five children, Georg — , Agnes — , Hermann — , Johanna — , and Marie — Although Liebig was Lutheran and Jettchen Catholic, their differences in religion appear to have been resolved amicably by bringing their sons up in the Lutheran religion and their daughters as Catholics. This decision actually worked to Liebig's advantage.
As an independent venture, he could ignore university rules and accept both matriculated and unmatriculated students. From to , the laboratory was housed in the guardroom of a disused barracks on the edge of town. An open colonnade outside could be used for dangerous reactions. Liebig could work there with eight or nine students at a time. He lived in a cramped apartment on the floor above with his wife and children.
Liebig's students were from many of the German states, as well as Britain and the United States, and they helped create an international reputation for their Doktorvater.
His laboratory became renowned as a model institution for the teaching of practical chemistry. The new chemistry laboratory featured innovative glass-fronted fume cupboards and venting chimneys. It involved an ingenious array of five glass bulbs, called a Kaliapparat to trap the oxidation product of the carbon in the sample, following combustion of the sample.
Before reaching the Kaliapparat, the combustion gases were conducted through a tube of hygroscopic calcium chloride , which absorbed and retained the oxidation product of the hydrogen of the sample, namely water vapor. Next, in the Kaliapparat, carbon dioxide was absorbed in a potassium hydroxide solution in the three lower bulbs, and used to measure the weight of carbon in the sample.
A charcoal furnace a sheet-steel tray in which the combustion tube was laid was used for the combustion. Poisonnier in , and by Finnish chemist Johan Gadolin inDuplicate dummy configurations were then pared from the exhaustive list to obtain a unique set of In this case, the data point below the terminal circle and all data points on a horizontal ray to its right are grouped into a horizontal leg of a square isoquant. This hypothesis l treated in historical detail by Paris, implies that the agricultural production function should have isoquants with square comers located on the expansion path. Since all isoquants have "comers," any isoquant will intersect an isocost line with strictly positive prices at the comer of the isoquant. He received a small stipend, without laboratory funding or access to facilities.
It involved an ingenious array of five glass bulbs, called a Kaliapparat to trap the oxidation product of the carbon in the sample, following combustion of the sample.
Next, in the Kaliapparat, carbon dioxide was absorbed in a potassium hydroxide solution in the three lower bulbs, and used to measure the weight of carbon in the sample.