O problema técnico-econômico da adubação

Zagatto, A. G.; Gomes, F. Pimentel

doi:10.1590/S0071-12761960000100016

Resumo

The authors discuss from the economic point of view the use of a few functions intended to represent the yield y corresponding to a level xof the nutrient. They point out that under conditions of scarce capital what is actually most important is not to obtain the highest profit per hectare but the highest return per cruzeiro spent, so that we should maximize the function z = _R - C_ = _R_ - 1 , C C where R is the gross income and C the cost of production (fixed plus variable, both per hectare). Being C = M + rx, with r the unit price of the nutrient and Af the fixed cost of the crop, wo are led to the equation (M + rx)R' - rR = 0. With R = k + sx + tx², this gives a solution Xo = - Mt - √ M²t² - r t(Ms - Kr)- _____________________ rt on the other hand, with R = PyA [1 - 10-c(x + b)], x0 will be the root of equation (M + rx)cL 10 + r 10c(x + b) = 0 (12). Another solution, pointed out by PESEK and HEADY, is to maximize the function z = sx + tx² _________ m + rx where the numerator is the additional income due to the nutrient, and m is the fixed cost of fertilization. This leads to a solution x+ = - mt - √m²t² - mrst (13) _________________ rt However, we must have x+< _r_-_s_ I if we want to satisfy t _dy_ > r. dx This condition is satisfied only if we have m < _(s__-__r)² (14), - 4 t a restriction apparently not perceived by PESEK and HEADY. A similar reasoning using Mitscherlich's law leads to equation (mcL 10 + r) + cr(L 10)x - r 10cx = 0 (15), with a similar restriction. As an example, data of VIEGAS referring to fertilization of corn (maize) gave the equation y - 1534 + 22.99 x - 0. 1069 x², with x in kg/ha of the cereal. With the prices of Cr$ 5.00 per kilo of maize, Cr$ 26.00 per kilo of P2O3,. and M = Cr$ 5,000.00, we obtain x0 = 61 kg/ha of P(2)0(5). A similar reasoning using Mitscherlich's law leads to x0 = 53 kg/ha. Now, if we take in account only the fixed cost of fertilization m = Cr$ 600.00 per hectare, we obtain from (13) x+ = 51 kg/ha of P2O5, while (14) gives x+ - 41 kg/ha. Note that if m = Cr$ 5,000.00, we obtain by formula (13) x+ = 88 kg/ha of P2O5, a solution which is not valid, since condition (14) is not satisfied.

Escola Superior de Agricultura «Luiz de Queiroz»

SUMMARY

The authors discuss from the economic point of view the use of a few functions intended to represent the yield y corresponding to a level xof the nutrient. They point out that under conditions of scarce capital what is actually most important is not to obtain the highest profit per hectare but the highest return per cruzeiro spent, so that we should maximize the function

z = _R - C_ = _R_ - 1 ,

C C

where R is the gross income and C the cost of production (fixed plus variable, both per hectare). Being C = M + rx, with r the unit price of the nutrient and Af the fixed cost of the crop, wo are led to the equation

(M + rx)R' - rR = 0.

With R = k + sx + tx², this gives a solution

X_o = - Mt - √ M²t² - r^t(Ms - Kr)-

_____________________

rt

on the other hand, with R = P_yA [1 - 10^-c(x + ^b)], x₀ will be the root of equation

(M + rx)cL 10 + r 10^c(x + ^b) = 0 (12).

Another solution, pointed out by PESEK and HEADY, is to maximize the function

z = sx + tx²

_________

m + rx

where the numerator is the additional income due to the nutrient, and m is the fixed cost of fertilization. This leads to a solution

x⁺ = - mt - √m²t² - mrst (13)

^{_________________}

rt

However, we must have x⁺< _r_-_s_ I if we want to satisfy

t

_dy_ > r.

dx

This condition is satisfied only if we have

m < _(s__-__r)² (14),

- 4 t

a restriction apparently not perceived by PESEK and HEADY. A similar reasoning using Mitscherlich's law leads to equation

(mcL 10 + r) + cr(L 10)x - r 10^cx = 0 (15),

with a similar restriction.

As an example, data of VIEGAS referring to fertilization of corn (maize) gave the equation

y - 1534 + 22.99 x - 0. 1069 x²,

with x in kg/ha of the cereal. With the prices of Cr$ 5.00 per kilo of maize, Cr$ 26.00 per kilo of P₂O₃,. and M = Cr$ 5,000.00, we obtain x₀ = 61 kg/ha of P₂0₅. A similar reasoning using Mitscherlich's law leads to x₀ = 53 kg/ha.

Now, if we take in account only the fixed cost of fertilization m = Cr$ 600.00 per hectare, we obtain from (13) x+ = 51 kg/ha of P₂O₅, while (14) gives x+ - 41 kg/ha.

Note that if m = Cr$ 5,000.00, we obtain by formula (13) x+ = 88 kg/ha of P₂O₅, a solution which is not valid, since condition (14) is not satisfied.

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LITERATURA CITADA

Recebido para publicação em 19/9/1960.

FRENCH, B. L. - 1956 - Functional Relationships for Irrigated Corn Response to Nitrogen. Journal of Farm Economics 38: 736 - 747.
HEADY, E. O. - 1952 - Economics of Agricultural Production and Resource Use. Prentice Hall, New York.
PESEK, J. e E. O. HEADY - 1958 - Derivation and Application of a Method for Determining Minimum Recommended Rates of Fertilization. Soil Science Society of America Procedings 22: 419 - 423.
PIMENTEL GOMES, F. - 1953 - The Use of Mitscherlich's Regression Law in the Analysis of Experiments with Fertilizers. Biometrics 9 : 498 - 516.
PIMENTEL GOMES, F. e CLOVIS P. ABREU - 1959 - Sôbre uma Fórmula para o Cálculo da Dose Mais Econômica de Adubo. - Anais E. S. A. «Luiz de Queiroz» 16: 191-198.
SPILLMAN, W. J. - 1924 - The Law of Diminishing Returms. World Book Company. New York.
STEVENS, W. L. - 1951 - Asymptotic Regression. Biometrics 7: 247 - 267.
VIEGAS, G P. - 1954 - 55 - Adubação do Milho II. Bragantia 14: 149 - 170.