Friday, August 11, 2017

Following up - How Uncertainty Impacts Manure Application Rate Decision Making

I received a question the other day about my blog post from mid-July dealing with uncertainty in manure application rate selection. If you recall, I talked about the concept of insurance Nitrogen (N) and why it wasn’t advisable. That’s where I said something a little controversial, with uncertainty in inputs to our decision, such as manure nitrogen content, the ideal nitrogen rate goes down (just a little bit, but still down). This week I was asked how I made that conclusion and if there was different uncertainty from different types of manure.

Since the second part is a little easier, I’ll start with that one: Is there uncertainty for different types of manure (swine, solid cattle, liquid dairy) or are they all equally variable? The data I have suggests the coefficient of variation (standard deviation divided by the average) in nitrogen content of hog, dairy, layer, and beef manures were about 23, 40, 30, and 33%.

Keep in mind this only focuses on uncertainty in the actual nutrient content of the manure – there are actually numerous things that cause uncertainty in what we actually apply. Things like the amount of volatilization that occurs during agitation and application, the availability of nitrogen in the manure, how accurately we are hitting the desired application rate, how uniformly we are applying manure, and even when we apply the manure relative to crop need. Some of these we can control, some of these we can reduce the impact of by picking the right type of equipment, and in some cases we can get a better idea through sampling, but there will always be some uncertainty remaining (You can read more about dealing with uncertainty here, the value of manure sampling here, and about rate control here I know I hit on these 4R topics a lot, but while achieving the Iowa Nutrient Reduction Strategy is going to require more than just the 4R’s, they are R’s that impact the value we get from our manure helping make manure a more valuable commodity for our livestock producers.

Now the hard part of this question, why did I say uncertainty lowered the ideal manure application rate? The simple answer I created a model, liked the assumptions I made, and believed it. However, I recognize that is a very unsatisfactory answer to a curious farmer, a consultant that wants to give better information to a farmer, and even this extension engineer who is trying to provide the answer. So I looked back at my work and thought, there has to be an easier way to say this is true. Then in that moment a light bulb, I should be able to do this graphically (and who doesn’t like a picture!).

So a little bit to help to get us started, I need to go over two pictures then we’ll work on putting them together (excuse my lack of graphical design). The first is the Maximum Return to Nitrogen graph; the one shown in figure 1 is for Southeast Iowa for corn in a corn-soybean rotation with estimated nitrogen prices of $0.30 a pound and a value of corn at $3.50 per bushel. In case you haven’t seen the Maximum Return to Nitrogen Tool before you can find it here, but briefly its six states’ (Illinois, Iowa, Michigan, Minnesota, Ohio, and Wisconsin) approach to recommending nitrogen application rates. There are three lines plotted on the curve: the first (blue) is the gross return to nitrogen – this is the price of corn per bushel times the estimated number of bushel we’d expect from different nitrogen application rates (this is based on years of data and numerous research plots in those regions). The second (green) is the cost of fertilizer, it is the fertilizer price multiplied by the number of units of N we applied. Finally, we get the third line (red) which is the difference between the gross return and a fertilizer cost. This line shows us the point (the highlighted diamond) where that unit of nitrogen resulted in just enough corn yield to cover the cost of buying that last bit of nitrogen.

 Figure 1. Maximum return to nitrogen for corn in a corn-soybean rotation for Southeastern Iowa.

The second picture I’m going to show you is a normal distribution. I know what you are thinking the same thing my dad, Mr. Manure, would say, “That’s Greek to me.”(now you know where these terrible jokes come from!). Essentially there are a few key points from this figure we need to know. If you were told to guess one value to pick on this curve you’d want to pick the one at the peak, or the mean (its where you’d be most likely to be right) denoted with the Greek letter µ. The next is the shape of this curve (how wide or narrow that bell curve looks) is controlled by the standard deviation, denoted with the Greek letter σ. A big standard deviation leads to short, wide curves and while small standard deviations lead to narrow, high peaked curves. This standard deviation gives us our uncertainty – you see, about 68% of the time we’ll be within one standard deviation of the mean (the light blue area in figure 2), 95% of the time we’ll be within two standard deviations (the light blue and the dark blue in figure 2), and 99.7% of the time we’ll be within three standard deviations.

Figure 2. Illustration of a normal distribution.

Now the fun part – the math I did to get my original answer. I simply put this normal distribution onto our Net Return to N curve and looked for the point on the net return curve where the area under the net return to N curve of the left-half of the normal distribution and the right-half of the normal distribution are equal (the math intensive part is they have to balance when weighting this area by the probability of that N application rate occurring).

Figure 3. Normal distribution overlaid onto the maximum return to N curve (for most of Iowa, corn in a corn-soybean rotation).

What the uncertainty does is basically flatten out the yield curve. Say you are trying to put on 140 lb N/acre. There is a chance you hit this rate, there is also a chance that you only put on 100 lb N/acre and similarly there is a chance you actually put on 180 lb N/acre.

If you think about what happens at low application rates (below MRTN), there is a chance that there was more N in the manure than you anticipated. This would work to your advantage because we are in an extremely sensitive part of the yield response curve where just a little more nitrogen results in a lot more yield. If you think about, this is exactly what MRTN was designed to say; if we are below MRTN the benefit of a little extra N is worth it.

If you are on the right side of the curve (above MRTN) it works the opposite way. A little less N only causes minimal yield drop because the yield curve is relatively flat here, but a little more wastes the value of the nitrogen and gives you essentially no yield gain. Again, this makes sense because it is effectively what MRTN was designed to tell you.

Also, it’s reasonable any uncertainty we have is going to make us less profitable than if we know perfectly what we put on. It’s like playing the lottery, if we knew the winning number beforehand we could be sure to pick it every time. Even if we were trying for the right amount there is some probability that we missed that “ideal rate” and as a result either over applied nitrogen or under applied just a little.

Why did I say it was slightly more profitable to under apply than over apply? The difference between the two rates is marginal (about 5-10 pounds an acre). What I think is that as you get close to the MRTN rate even though you are a little below MRTN, the yield curve has already flattened out quite a bit. This puts us in a situation where if we are just a little low it doesn’t really change our yield, but if we are over that amount it’s costing us $0.30/lb N that could have been used elsewhere. I tried to show you why this would happen pictorially in figure 3). If you place a normal distribution over the Net Return to N (with the red dot representing the ideal point), you’ll see that the net return fell off a bit more quickly on the right side of the distribution then on than it rose on the left side of the distribution. This would mean if we have uncertainty, our ideal rate is actually to the left of the MRTN with no uncertainty.

However, you’ll also note the shape of the distribution matters. If it was narrower, the MRTN with and without uncertainty might be the same point. If the uncertainty is bigger, it might even move us to the right, that is a higher application rate than MRTN. This would happen if our distribution is wide enough that our uncertainty curve (that normal distribution) encompasses part of the yield response curve where yield responds rapidly to increases in nitrogen (figure 4).

Figure 4. Maximum return to nitrogen for corn in a corn-soybean rotation for Southeastern Iowa with normal distribution with high uncertainty overlaid on the curve.