The ‘Dairy N Fertiliser Advisor’ - a tool
to predict optimal N application rates
in grazed dairy pastures
Agricultural Research Division Technical Report
Author: Kerry Stott, Bill Malcolm, Cameron Gourley
Project CMI Number: 104219
© The State of Victoria Department of Economic Development, Jobs, Transport and Resources, May 2015
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Contents
Executive Summary
1
Introduction
2
Method
2
The economic framework
Response function parameters
Calculations embedded in the N-Advisor
Farmer input to the N-Advisor
2
4
6
6
Results and Discussion
7
Bibliography
10
Executive Summary
The Dairy Nitrogen Fertiliser Advisor (the ‘N-Advisor’) is an approach to analysing decisions about nitrogen (N)
applications to pastures that enables dairy farmers, along with their advisors, to examine the profitability of N fertiliser
applications to a particular paddock for a particular grazing rotation. The approach mimics the steps that decision-makers
currently go through when deciding how much N to apply, and adds a rigorously defined estimate of the dry matter
production that can be expected from the range of possible applications of N.
The N-Advisor captures on a regional and seasonal basis the pasture yield responses to N fertiliser derived from 65 N
fertiliser experiments undertaken across Australia over the past 40 years (which equates to nearly 6,000 Nitrogen
fertiliser v pasture response data sets), and calibrates these to expected pasture consumption for the prevailing
conditions facing individual farmers. Based on production economic theory, N fertiliser recommendations developed by
using the N-Advisor take into account not only the response function calibrated by the decision-maker to the area of
pasture to which the N is to be applied, but also the costs of the fertiliser and the value of the extra pasture consumed.
The N-Advisor allows users to perform what-if analyses, such as exploring the effect on the profit maximising level of N
of changing the capital cost of N fertiliser applied, or changing the value of the dry matter (DM) consumed. The NAdvisor also enables risk associated with production outcomes to be taken into account.
The N-Advisor provides a firm foundation for extension to the target population of farmers and advisors. Opportunity for
subsequent additional capability also exists.
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1
Introduction
A ‘Generalised Model of N Fertiliser Responses’ developed by Chia and Hannah (2013) has been incorporated into into
an economic framework, and embedded in a web based decision support tool called the ‘Dairy Nitrogen Fertiliser
Advisor’ (N-Advisor). The aim is to help dairy farmers decide how much nitrogen to apply to a particular paddock for a
particular grazing rotation.
The N-Advisor is an output of the ‘Dairy Nitrogen for Greater Profit’ project funded by the Gardiner Foundation in
response to the increasing challenges being faced by dairy farmers to use N fertiliser efficiently and to reduce potentially
adverse environmental impacts through over-application of N fertiliser. The N-Advisor is accessible at
http://vro.depi.vic.gov.au/dpi/vro/vrosite.nsf/pages/nitrogen-advisor.
Method
The economic framework
The economic framework underpinning the N-Advisor is that of production economics. From economic theory, the
decision rule to maximize profit from using a variable input such as N, with all other inputs held constant, with full
information and without any capital constraint, is to apply N up until the extra return from an extra kilogram of N applied
just exceeds the additional cost of the extra kilogram of N applied.
The total product curve (or response function) in Figure 1 shows the way an output, such as pasture in the current
production period, responds to inputs such as N fertiliser. The response function follows the operation of the law of
diminishing marginal returns and input applications define three ranges of production:

Increasing marginal returns. In this range of input use, the initial units of an input have the effect of increasing
quantities of output with each additional unit of input added. Total production increases at an increasing rate.

Diminishing marginal returns. In this range of input use, when extra inputs are added to production, the successive
extra units of production that result decline in magnitude. Total production continues to rise, at a diminishing rate.

Negative returns. In this range of input use, as more inputs are added, each extra unit of input reduces total
production. The same total production can be achieved using less total inputs.
The total product curve tells farmers a lot about how much input to add to boost production. In Figure 1 the range of
input use where extra inputs add increasing amounts of output per unit, the increasing returns stage of production, is the
‘too little’ level of input. Obviously it makes no sense to use more inputs to make less product, even if the input is free.
The range of negative returns is the ‘too much’ level of inputs. Regardless of the cost of the input, farmers trying to
maximize profits would need to use inputs at least up to the level where the extra production that results from an extra
unit of input has reached a maximum and is starting to decline. This is the level of variable input use where average
product is at a maximum. Up to this level of input use the extra product that results is pulling up the average output from
all the previous applications of the variable input. In between the ‘too little’ level of input and the ‘too much’ level of input
is the ‘just right’, profit maximizing level of variable input use.
The precise ‘best level’ of variable input to use is where the value of the last extra unit of production brings in just a little
more than the cost of the last unit of input, e.g. where the last unit of fertiliser used costs $1 and adds extra output that
adds $1.01 to income. This last unit of input increases total product and adds $0.01 to profit. All the previous units of
input added more than this amount to total profit.
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2
Figure 1. Response function: inputs to output (Malcolm et al. 2005 p69)
The most profitable level of N to use is where marginal revenue (MR) of output (𝑦) from the added input equals the
marginal cost (MC) of the added input (𝑁). Marginal revenue is the price of the output (Py) multiplied by the marginal
product (MPn). Marginal cost is the cost of the extra input applied (𝑃𝑛 ), so Py*MPn=𝑃𝑛 * 𝑁. In algebraic terms this equation
can be rearranged to: MPn = Py/Pn. This is the same as identifying the level of variable input use where the slope of the
response function (MPn) equals the ratio of the cost of the input (N fertiliser applied) to the value of the output, which in
the case of pasture grazed for dairying is the the dry matter (DM) consumed (Bishop and Toussaint 1958, p 46).
Fertiliser cost is ‘as spread’, including interest cost of financing the input, and the extra pasture consumed is valued using
the cost of obtaining an equivalent quantity of a substitute source of metabolisable energy (ME). The profit maximising
amount of N applied decreases with the increasing cost of N fertiliser and increases with the increasing value of the DM
consumed. To account for risk and uncertainty in production outcomes, the N-Advisor allows for variation around the
‘best bet’ level of extra pasture produced and consumed.
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Response function parameters
The generalised experimental response function
The generalised experimental response function developed by Chia and Hannah (2013) relates N application rates to
pasture consumption in a particular paddock of pasture, at a particular time of the year, in a particular region. The
response function shows diminishing returns up to 100 kg N/ha and pasture DM consumption is scaled as a proportion of
obtainable yield. The N response depends on the Australian state and season to which the experimental data relates
(Figure 2).
Figure 2. Predicted response curves for proportional DM yield and nitrogen fertiliser rate for states and seasons.
The generalised N response functions for regions and seasons were developed using data compiled from fertiliser trials
carried out in Australia between 1955 and 2009 with the majority of the experiments conducted during the 1960s and
1970s. This involved gathering and standardising data from 65 identified N fertiliser experiments conducted across
Australia. These experiments contributed to approximately 6000 N fertiliser – pasture response data sets.
Chia and Hannah analysed the data and meta data (such as location, dates, soil type etc.) on pasture yield response to
N fertiliser and developed a quantitative non-linear mixed effects model to describe this response. The model was based
upon the exponential Mitscherlich function and included mixed effects models for the coefficients. These comprised fixed
effects for Australian State and season (spring, summer, autimn, winter), soil phosphorus status (limiting or non-limiting)
and harvest type (initial or residual), and nested random effects for location and partition (where ‘partition’ refers to a
single trial or sub-section of a trial).
The analysis was hampered by the extent of the data, patchy availability of meta data, only two nitrogen rates applied in
the majority of trials, skewed representation of states, regions and times, and selection biases arising from trial protocols.
Despite the limits of the national data set, the generalised model usefully predicts pasture response to applied N as a
proportion of obtainable yield.
Mathematically, the generalised N response function can be represented as follows:
𝑦 = 𝛼(1 − 𝑒 −𝛽−𝜆𝑁 ) + 𝜖
(1)
where,
𝑦 is output from the added input N
𝛼 is the maximum attainable yield when N applied is unlimiting and has had sufficient time to express itself,
𝛽 is an implicit measure of existing soil N,
𝜆 is a constant and it is a measure of the curvature of the response function, and
𝜖 is the error term.
All parameter values used in the N-Advisor are the values estimated by Chia and Hanna (2013) and are for non-residual
(initial) yields and non-limiting soil phosphorus. As modelled, 𝛼 is a proportional response, and must be calibrated to the
The Dairy Nitrogen Fertiliser Advisor
4
farm paddock by a farmer and/or advisor using their judgement to form a subjective estimate of yield at a defined level of
N application. The values for 𝛽 vary with the fixed effects of state and season, and are shown in Table 1. Finally, the
value of 𝜆 is 0.026 for all locations and seasons.
Table 1. 𝜷 values for the generalised experimental response function
Region
Season
β values
New South Wales
New South Wales
New South Wales
Queensland
South Australia
Tasmania
Tasmania
Tasmania
Victoria
Victoria
Victoria
Western Australia
Western Australia
Autumn
Spring
Winter
Summer
Autumn
Autumn
Spring
Winter
Autumn
Spring
Winter
Autumn
Winter
1.200
0.680
0.400
0.250
0.440
0.880
0.740
0.990
1.100
0.630
1.000
1.100
1.400
Calibrating the generalised experimental response function to a particular paddock
According to Chia and Hannah (2013 p16), the “values for 𝛼, for purposes of real-time, on-site prediction may be
[inferred from] an estimated zero-applied N dry matter yield, 𝑦0 , as follows:
𝜶 = 𝒚𝟎 ⁄(𝟏 − 𝒆−𝜷 )
(2)
where, 𝒚𝟎 is the expected pasture consumption (kg DM/ha) at 0 N fertiliser.”
This method is encapsulated in the N-Advisor. As indicated in Figure 3, to determine 𝛼 from any applied N dry matter
yield, farmers and/or their advisors estimate either the level of DM/ha that would be produced in the paddock that is
about to be fertilized if a defined quantity of extra N/ha was applied in the time period and region described by the
generalised N response function, or, users may provide an estimate of the extra DM/ha that would be produced with zero
N/ha applied (𝑦0 in Figure 3) in the time period and region described by the generalised response function.
Figure 3. Pasture response curve to applied nitrogen fertiliser.
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5
Calculations embedded in the N-Advisor
A number of measures are derived from the calibrated response function:

The pasture consumed from the last unit (kg) of N applied (the Marginal Product) is obtained from its derivative
or slope (equation 3).
𝑀𝑃 = 𝛼𝜆𝑒 (−𝛽−𝜆𝑁)

(3)
The profit maximising application rate of N is obtained by equating the derivative of the response function
(equation 4) to the ratio of the cost of the input (𝑃𝑛 ) to the value of the output (𝑃𝑦 ) and solving for N (equation 5).
𝛼𝜆𝑒 (−𝛽−𝜆𝑁) = 𝑃𝑛 ⁄𝑃𝑦
(4)
⇒ 𝑁 ∗ = (1⁄−𝜆)(ln(( 𝑃𝑛 ⁄𝑃𝑦 ) ⁄(𝛼𝜆)) + 𝛽)
(5)
where N* is the profit maximising N.

Pasture consumption at the profit maximising N is calculated by substituting N* for N in the response function
(equation 6).
∗
𝑦 = 𝛼(1 − 𝑒 −𝛽−𝜆𝑁 )

(6)
The Marginal Revenue (MR) (expressed as a decimal) on the money invested in the last unit of N applied is
calculated as shown in equation 7.
𝑀𝑅 = ((𝛼𝜆𝑒 (−𝛽−𝜆𝑁) 𝑃𝑦 ) − 𝑃𝑛 ) / 𝑃𝑛
(7)
Farmer input to the N-Advisor
Using the N-Advisor, and to calibrate the generalised experimental response function to the expected absolute level of
pasture consumption in a particular paddock of pasture for the particular time of year (season) and region (state),
farmers and/or their advisors are asked to provide:
1.
the most likely post-grazing dry matter mass in the area to be fertilized
2.
the most likely pre-grazing dry matter mass in the area to be fertilized
3.
the N rate (zero or non-zero) judged most likely to achieve these outcomes
Answers to these three questions calibrate the position of the generalised N response function to the area of pasture that
is the subject of the fertiliser decision. Expected pasture consumption is determined by subtracting the expected postgrazing dry matter mass from the pre-grazing dry matter mass. Users are also asked to provide estimates of the 'as
spread' cost of N fertiliser and the market value of the extra pasture consumed, based on the cost of obtaining an
equivalent quantity of ME from a substitute source. This cost is influenced by (i) the season and the supply of sources of
ME at this time and (ii) the value of the product for which the ME is to be used to produce at this time, i.e. the milk price
at the time of year the decision to add N is being made.
A simple, robust way of valuing the extra pasture consumed by the grazing animals as a result of the added N is to use
market opportunity costs and replacement values. From a profit maximising perspective, where pasture is produced and
used in a farm system its value must be greater than what it would return if it was sold for agistment or fodder
conservation. Furthermore, its value must be less than the value of an equivalent substitute input to the system
(Johnson and Hardin 1955). If the pasture was worth more than the substitute input, then the substitute input would be
used and the pasture would not be used in the farm system. Thus pasture used in a farm system has a value that lies
between agistment or fodder conservation value and the cost of an equivalent substitute input such as ME from barley, at
the time of year the N fertiliser decision is being made.
The Dairy Nitrogen Fertiliser Advisor
6
During years or seasons (summer, early autumn) of relative pasture shortages it is appropriate to use the maximum
value based on the value of substitute ME at that time. During years or seasons (spring) of relative abundance in
pasture DM it is appropriate to use the minimum value based on the agisting out or the net return that could be realised
by baling it up and selling it at that time. The difference between the higher price of substitute ME when pasture is in
deficit, and the lower price when it is in surplus, is in large part due to the costs of conserving pasture, storage and
transport.
An examples of the calculation involved in valuing pasture using the market price of equivalent dietary nutrients in barley
is shown in Table 2. This table contains a full list of the input data required, a brief description, and some sample values.
Table 2. Input data and values
Variable
Description
Value
Season
Pasture response to N varies throughout the growing season. Select from spring,
summer, autumn or winter.
Pasture response to N varies from location to location. Select from NSW,
Queensland, South Australia, Tasmania, Victoria or Western Australia
Select the most likely residual mass following grazing for the current rotation in the
range 1,100 to 1,500.
autumn
1,200
kg
DM/ha
Select the most likely pre-grazing mass for the prevailing conditions (soil
temperature and moisture) over the current rotation for your nominated N
application. Range is 1,500 to 3,000.
An amount between zero and 100kg /ha of N that you would usually apply to
achieve the nominated pre- and post-grazing mass. The quantity refers to
elemental nitrogen, not the quantity of fertiliser applied such as urea (46% N) or
sulphate of ammonia (21% N).
The N-equivalent total cost between $500 and $2,000/t N to have your fertiliser
delivered and spread. If using urea, divide the ‘as spread’ cost by 0.46 to obtain
the ‘as spread’ cost in Nitrogen equivalents. If using sulphate of ammonia, divide
costs by 0.21.
The marginal extra pasture consumed is valued using the market price for
substitute ME. Select a value in the range $100 to $500/t DM. If using feed
barley for comparison, multiply the current barley price (say $300/t DM
delivered(a)) by the ratio of the ME concentration in pasture (11.5 MJ/kg DM) (b) to
the ME concentration in barley (12.3 MJ/kg DM)(b).
2,500
kg
DM/ha
50
kg
N/ha
1500
$/t N
279
$/t DM
Region
Most likely
post-grazing
dry mass
Most likely pregrazing dry
mass
Usual Nitrogen
application
Nitrogen cost
‘as spread’
Market price of
pasture
consumed
Units
Victoria
(a)
Prices for feed stuffs delivered on-farm can be sourced from the Dairy Australia “Hay and Grain Market Report” available at:
http://www.dairyaustralia.com.au/Pastures-and-Feeding/Supplements-and-nutrition/Supplementary-feeds-2/National--Hay-GrainMarket-Report.aspx
(b)
Nutritive characteristics of supplementary feeds used in the Victorian Dairy Industry can be found at:
http://www.depi.vic.gov.au/agriculture-and-food/dairy/feeding-and-nutrition/dairy-supplement-list
Results and Discussion
The N-Advisor provides estimates in a graphical format of the profit maximising N for the most likely dry mass of pasture
consumed, and for better and worse pasture consumption outcomes (Figure 4). In practice the actual DM consumed that
will result from an application of N will differ from the best estimate that is made at the time of the decision. Actual
pasture DM consumed cannot be predicted with precision, except by chance. Even if the response function that is
applied to the paddock was accurately predicted, the resulting output that is consumed will depend on the extent and
timing of the subsequent rainfall and temperature events that will occur during the grazing season, as well as the
management of the animals at the time of grazing. This situation applies of course to N fertiliser decisions made with or
without using the N-Advisor. Thus an important part of the process of using the N-Advisor to inform the decisions farmers
make about applying N is for them to consider a reasonable range of the possible eventual DM consumed as a result of
The Dairy Nitrogen Fertiliser Advisor
7
the decision to apply a particular quantity of N. Then, the decion-maker is in a position to make a well informed ‘bet’. To
this end, the N-Advisor identifies the most profitable N application for the cases where the actual DM available to be
consumed is 20% above or below the most likely most likely level of pasture consumption.
Hovering the cursor over the text box in the N-Advisor will show the N applied, most likely DM consumed, pasture
consumed for the last kilogram of N applied (the marginal product), and the marginal return on the last dollar invested in
N expresses as a percentage. For the input values shown in Table 2, at the profit maximising N with the most likely
pasture consumption, these values are 32kg N/ha, 1,223 kg DM/ha, 5.4 kg DM/kgN and 0%, respectively.
The idea that maximum profit is earned when the last unit of input makes a profit of zero percent sometimes confuses,
causing people to wonder how is it that profit can be maximised when only a tiny return is made on the last unit of input.
The point is that by using other inputs previously up to this last input, the producer has earned bigger profits on all the
earlier inputs. Adding all the profits from each input gives total profit from all inputs.
Figure 4. Dairy N Fertiliser Advisor Interface.
The Dairy Nitrogen Fertiliser Advisor
8
The N-Advisor allows users to perform sensitivity ‘what-if’ analysis, such as exploring the effect of changes in the cost of
N fertiliser applied, or the value of the DM consumed. The value of the DM consumed can vary according to the time of
the year, be influenced by the cost of alternative sources of ME which in turn are influenced by the seasonal supply of
ME and the value of the end product for which it is used (i.e. the milk price). Quite large annual changes in the cost of N
fertiliser are also possible, such as in 2008 when the cost of natural gas (a key component in the manufacturing process)
spiked.
The baseline results shown in the text box in Figure 4 are highlighted in Table 3. Also shown in this Table are the profit
maximising N application rates for a 20% change in expected pasture consumption, the fertiliser cost and the value of
pasture consumed. The Table shows that the profit maximising amount of N applied increases not only as the expected
pasture consumption increases, but also as the cost of N fertiliser (including financing costs) decreases, and as the value
of the DM consumed increases. The magnitude of the percentage changes in the profit maximising N application is
similar to the percentage changes in these causal factors.
Table 3. Sensitivity analysis for profit maximising N application rates (percentage change from the 32 kgN/ha
baseline in brackets)
Profit maximising level of N (kg N/ha)
Fertiliser 'as
spread' cost
($/t N)
Value of
pasture
consumed ($/t
DM)
Price
ratio
Pasture consumption
20% worse than
expected (kg DM/ha)
Most likely pasture
consumption (kg
DM/ha)
Pasture consumption
20% better than
expected (kg DM/ha)
1,500
223
6.7
15
24 (-25%)
31
1,500
279
5.4
24 (-25%)
32
39 (+22%)
1,500
335
4.5
31
39 (+22%)
46
N application associated with a 10% return on capital invested (kg N/ha)
1,650
223
7.4
11
20
27
1,650
279
5.9
20
28 (-12%)
35
1,650
335
4.9
27
35
42
N application associated with a 20% return on capital invested (kg N/ha)
1,800
223
8.1
8
17
24
1,800
279
6.5
17
25 (-21%)
32
1,800
335
5.5
24
32
39
Note that the price ratio in column 3 is equivalent to the pasture consumed for the last kilogram of N applied (i.e. the
marginal product) as per Equation 4. The marginal product and the marginal return are the same for all three pasture
consumption scenarios at the point where N use is optimised (holding other variables constant). However, as shown in
the Figure 4 and Table 3, the profit maximising N application differs between the three scenarios in line with expected
pasture yield and consumption outcomes. Users can alter the expected pre- and post- grazing mass to examine how a
lower pasture utilisation would decrease the optimal N application in grazed dairy pastures.
The Dairy Nitrogen Fertiliser Advisor
9
Acknowledgements
We are grateful to Matt Cox and Oliver Lardner for the computer programming involved in developing the N-Advisor.
Bibliography
Bishop C E and Toussaint W D (1958). Agricultural Economic Analysis, John Wiley and Sons, New York.
Chia K. and Hannah M. (2013). A calibrated model for pasture yield response to nitrogenous fertiliser. Internal report of
‘Dairy Nitrogen for Greater Profit’ project. DEPI, Ellinbank
Johnson G.L. and Hardin L. (1955). ‘The Economics of Forage Evaluation’. Purdue Agricultural Experiment Station
Bulletin No. 623, Purdue University, West Lafayette, Indiana.
Malcolm B., Makeham J. and Wright V. (2005). The Farming Game, Agricultural Management and Marketing, 2nd edition,
Cambridge University Press, Melbourne.
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