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 http://cnrc.agron.iastate.edu/About.aspx, 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.

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.

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