Optimal input use when inputs affect price and yield

Input use has been shown to impact the variance of output and therefore risk. When inputs affect both output level and the price of the output, the variance of revenue and profit depend on each effect and their interactions. We analyze the effect of nitrogen (N) use rate applied to wheat on the variance of yield, revenue, and the price of wheat, when protein premiums (discounts) are applied. We find that N use rate increases the variance of yield, but reduces the variance of price. The net effect of N use rate on revenue and profit is variance increasing, but the variance effect is less than for yield alone. Optimal rates of N are about 60% higher with protein payments compared with a constant wheat price over all protein levels. Risk-averse producers apply less N than risk-neutral producers but, because revenue and profit risk is lower with protein payments, the reduction in N is less than if based on a constant price over all protein levels.