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Derivation and Application of a Method for Determining Minimum Recommended Rates of Fertilization¹

ABSTRACT

Where the cost of fertilizer applied in the field can be expressed as the cost of application plus the cost of fertilizer, it is important to determine the rate which gives the highest net return per dollar invested. This rate describes both the economic minimum rate of fertilizer application and the lower limit that should be included in making agronomic fertilizer recommendations.

If yield increase, Y₁, is expressed in the form of the quadratic equation, Y₁ = sx + tx², where x is the rate of fertilizer and s and t are constants; and the cost of fertilizer applied, Y₂, is given by Y₂ = m + rx, where m is the fixed application cost per unit area, and r is the price ratio of a unit of fertilizer to a unit of yield increase; then it is necessary to maximize the relationship
by setting d/dx (Y₁/Y₂ – 1) = 0 in order to derive the rate of fertilization that will produce the highest profit per dollar invested in fertilization.

A direct solution of the quadratic functional relationship above can be made and leads to
while a method of successive approximations may be used to solve relationships based on exponential functions of the Mitscherlich or Spillman form.

¹ Journal Paper No. 3301 of the Iowa Agr. and Home Econ. Exp. Sta., Ames, Iowa. Projects 1189 and 1293. Joint contribution of the Department of Agronomy and the Department of Economics and Sociology. Presented before Div. IV, Soil Science Society of America, Nov. 21, 1957, at Atlanta, Ga.

² Associate Professor of Soils and Professor of Agricultural Economics, respectively.

Received for publication January 10, 1958. Accepted for publication March 17, 1958.