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A Two Nutrient-Response Function with Determination of Economic Optima for the Rate and Grade of Fertilizer for Alfalfa¹

ABSTRACT

A replicated fractional factorial experiment with phosphorus and potassium fertilizer variables was conducted with alfalfa growing on Webster silty clay loam in 1952. Fertilizer was topdressed before growth was initiated in the spring, and the yields of hay determined from two cuttings. The range of fertilizer applications was wide enough to reach maximum yields.

The data were analyzed and three different two-variable equations were fitted to the observations. The equation which resulted in the best correlation with the observed data from this experiment was of the general form,
where Y is the yield, and K and P the pounds of K₂O and P₂O₅ per acre respectively. The correlation coefficient was 0.8793 and significant at the 1% level. The last four coefficients of K and P were also significant at this level, while the first coefficient of K was significant at the 5% level.

It was shown how this equation representing the response surface can be used to (1) determine the rates of P₂O₅ and K₂O combinations which resulted in the same yield levels; (2) calculate the least cost combination of the two nutrients for any given yield; (3) calculate the marginal yields of both nutrients; (4) determine the optimum level of application of each nutrient at constant levels of the other; and (5) calculate the optimum levels of both nutrients varying simultaneously.

Finally, it was illustrated how the least cost combination, and rates of application of the two nutrients varied as different prices for fertilizer elements and hay were assumed under the conditions of this experiment.

¹ Journal Paper No. J-2743 of the Iowa Agr. Exp. Sta., Ames, Iowa. Projects 1189 and 1135. The work was supported in part by a grant from the Tennessee Valley Authority. The authors wish to acknowledge the aid of Wm. Brown and Don Hunter. Contribution of the Departments of Agronomy and Agricultural Economics.

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

Received for publication April 8, 1955.