Introduction

The production of white corn is carried out in almost all geographical regions and rural areas of Mexico, in a great diversity of environments: wet, dry, cold and hot; it is also cultivated from sea level to about 3 400 masl, with a diversity of agricultural practices and uses of more than 80 indigenous ethnic groups and many mestizo peoples. In Mexico, there are about 64 races of corn (^{CONABIO, 2021}) resulting from a process of domestication, selection and evolution under cultivation carried out mainly by indigenous groups for 8 000 years and more recently also by mestizo groups (^{Wellhausen et al., 1951}; ^{Kato et al., 2009}, Biodiversidad, 2020, ^{SAGARPA, 2012}).

It is the grain par excellence, so it represented 88.3% of the national production of basic grains in the agricultural year 2020 (^{SADER-SIAP, 2020}). Which is produced under different production systems and with different technologies used that involve from small production units (PUs) of family self-consumption that allocate more than 80 percent of their production to their food, medium semi-commercial PUs that sell more than 63% of the grain, medium PUs of livestock self-consumption that allocate more than 83 percent of the corn they grow to feed the animals they raise and the large commercial PUs that sell more than 96% of their production to the national tortilla industry and that one that produces food for animal consumption, starches, vegetable oil and starches, among other products (^{Yunes and López, 2021}).

Despite the great wealth of biodiversity and the natural, cultural conditions and millenary experience of its production by ethnic groups, all the corn that Mexican society requires to meet its nutritional needs is not produced. In 2019, an amount equivalent to 38% of the apparent national consumption was imported and represented 59% of the national production (^{SEADER-SIAP, 2020}), although most of it was yellow corn, a raw material necessary to make food for livestock consumption and for agro-industry, which reflects a strong dependence on foreign supply.

Derived from production systems, fertilizer doses can be very varied, while some white corn producers in Mexico apply high doses of fertilizers and high sowing densities, with very mechanized processes to obtain high yields per hectare, to obtain good profits, on the other hand, there are producers with fewer resources, who apply lower doses of fertilization and therefore, lower yields but achieve higher profits per hectare because they use this input more efficiently. The hypothesis is that the former are using overdoses of inputs that lead them to obtain lower economic benefits per hectare.

This constitutes an economic problem, especially for those who sell the grain and for whom an economic solution can be shown. Producers applying doses below the economic optimum could increase their profits if they used the optimal dose on a smaller area according to their resource availability. The objective pursued in this research is to identify the optimal economic doses of fertilizer in field experiments, which will allow corn producers to use their scarce resources more efficiently to achieve the maximum return on investment in fertilizers.

Methodology

To address this study, the use of mathematical functions that empirically represent the law of diminishing returns by using mathematical functions mainly of exponential type, third- and second-degree polynomials, among others, is first analyzed (^{Martínez, 1972}; ^{Beattie and Taylor, 1993}, ^{Nicholson, 2002}). To demonstrate this, as a first point, the relationship between nutrients and yield is established (^{Martínez, 1972}; ^{Jauregui and Sain, 1992}; ^{Portillo, 2015}), which is expressed mathematically as follows: Let be the dose of nutrients (inputs) N, P, K, …, applied to the soil per unit area, denoted by the letters n, p, k, …, and Y the yield of the crop per unit area.

Y= f (n, p, k,) (1).

Where f can be set by a cubic, quadratic, exponential function, etc. Relationship (1) will be denoted as ‘production function’.

From the economic point of view, the application of N, P, K, …, has a cost that is determined by the relationship:

TC= g (n, p, k, ...) (2).

Where TC is the total cost of applying nutrients per unit area, which is a function of the doses of n, p, k, ..., relationship (2) will be denoted as ‘total cost function’. One of the most used functions in practice is a function of the form: TC= FC + P_N n + P_P p + P_K k +…, (3). Where: FC represents the fixed costs of application and the symbols P_N , P_P, P_K, …, the unit market prices of nutrients N, P, K, ..., respectively. The total income of the producer is given by P_YY. Donde: P_Y is the unit market price of the product.

Defining the economic optimum as the amount of nutrient that generates the yield or production of maximum profit or economic benefit, that amount of input is obtained, equalizing the value of the marginal product of the input (VMgPI= MgPI * P_YY) to the price of the input P_I= P_N, P_P, P_K, …, which is equivalent to equalizing the marginal product of the input (resource), (MgPI= ∂Y/∂I) to the price ratio of the nutrient to the price of the product (P_I / P_Y), as stated in equation (4). Mathematically, the problem is limited to finding the maximum of the product P_YY subject to constraint (3), assuming a linear cost function. Forming the following Lagrange function: L= P_YY + λ (TC - FC - P_N n - P_P p - P_K k - …).

Deriving L successively with respect to n, p, k, ..., and equalizing the corresponding equalities to the input/product price ratio, one obtains:

The system of equations (4) is solved simultaneously. The values of n, p, k, …, that are obtained must satisfy the system and are the optimal doses of resources. When considering the FC of the application of fertilizers, it may happen that the value of the increase in yield is not sufficient to cover the TC of the application. Low-dose applications may be insufficient to pay for FC, even if they cover the variable costs P_N n + P_P p + P_K k +…, of equation (3).

From the above, it is deduced that there must be a minimum value of n, p, k, ..., whose applications below these minimum doses will produce losses instead of the desired profit. Based on Pesek and Heady (1958) cited in (^{Martínez, 1972}), the quantities of inputs that make maximum net profits per monetary unit are considered as minimum doses of application. Considering that the total profit is the difference: π = P_YY - TC; from where the net profit per monetary unit spent is denoted as R and defined as follows: R = (P_YY - TC) / TC. The doses of N, P, K, ..., for which R is a maximum are calculated by simultaneously solving the system:

Note that TC in the above system is not a constant, but that TC is a function of doses n, p, k, .... The quantities obtained when solving the system (5) establish the minimum quantities that are required to obtain the highest profits, that is, the producer must fertilize with the minimum recommended amounts the area covered by these conditions and the rest that the minimum level of fertilization cannot cover must be left unsown (^{Martínez, 1972}; ^{Portillo, 2015}), since a different combination of dose and area will reduce the profits for a certain quantity of fertilizer.

This is explained because the classic response of crops to successive fertilizer additions follows the Law of diminishing ‘marginal returns’ ‘... at low doses of nutrients, increases in doses produce greater increases in yields, and conversely at high doses of nutrients, increases in doses produce minor increases in yields’ (^{Martínez, 1972}). So mathematical functions that empirically represent the law of diminishing returns are used.

However, second-degree polynomials allow for a very reasonable representation of a crop’s responses to nutrient applications. This second-degree polynomial allows calculating the economic optimum and the technical optimum due to the linear and quadratic effects that are observed between the nutrients and the yield of a crop. For the case of 1 and 2 two inputs, there are the following equations:

The system of equations (4) simply reduces to the following equation, in the case of a nutrient or a balanced fertilizer formula, (F):

Where:

Where: n* or f* is the optimal dose of N or F, respectively.

Results and discussion

The results of the soil analysis found low content of organic matter and inorganic nitrogen (N), while excess amounts of phosphorus (P₂O₅) and potassium (K₂O). In micronutrient content, there was Cu deficiency, low Fe and Zn level, medium CaO, high Mn and excess Mg and B. CEC was also low and medium pH (Table 1). Taking into account this information and the requirements of the corn to produce an expected average yield of 10 t ha^-1, the balanced fertilizer formula (F) was made with the following contents: 23% of N, 0% of P₂O₅, 26.53% of K₂O, 17.73% of CaO plus the mixture of micronutrients mentioned above.

Once the required fertilizer formula was determined, the experimental design was established, which was bifactorial treatments 2x5. The first factor consisted of two varieties of corn (Estrella and Celeste). The second factor in five fertilization doses (T0= absolute control; T1= producer’s dose; T2=1/2 of the optimal dose; T3= optimal dose and T4:1+1/2 of the optimal dose). T0 consisted of not applying any type of fertilizer to the soil or the plant, T1 was that applied by the producers of the region (400 kg ha^-1 of F) and the optimal dose was carried out based on the soil analysis, which was 1 200 kg ha^-1 (with 276N-0P-318K-213CaO + micronutrients) and served as a reference point for the other treatments.

Conclusions

Of the 4 treatments evaluated, the one that had the best response in the yield variable evaluated was T4 (the highest dose of fertilization), so in this aspect it was the best. While for this same variable, the control treatment had the lowest values. With the Celeste variety and the highest dose, the highest yield was obtained, which was 13.83 t ha^-1, while with the Estrella variety and this same fertilization dose, a yield very close to the previous one of 13.77 t ha^-1 was recorded. The estimated quadratic production function of white corn allows obtaining the optimal economic dose or maximum profit of F (848 kg ha^-1) and the optimal technical dose or maximum yield (1893 kg ha^-1).

The results obtained can serve as an economic criterion for making production decisions by the small producers of white corn, who have fewer resources to buy fertilizers, so that, optimally using this resource, they could apply only 424 kg of F in half a hectare and thus obtain the maximum net profit per peso spent on F, as well as by medium and large producers, who have more resources to acquire the amount of F that allows them to achieve the maximum profit. It is recommended that in future research the region to be explored be expanded, in such a way that the size allows to observe the type of classic response. It is also convenient to explore an experimental design of random blocks.