NFREC, Quincy Research Report 909
Maximum Economic Yield
A Mathematical Definition
Florida Agricultural Experiment Stations
Institute of Food and Agricultural Sciences
University of Florida, Gainesville
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IFAS
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Maximum Economic Yield: A Mathematical Definition
Maximum economic yield (MEY) of a crop results in maximum
profit per acre. Profit is determined by yield response to
production inputs, cost of inputs, and price of output. It has
been suggested that maximum economic yield is about 95% of maximum
yield, however, high inputcosts and low commodity prices may
reduce MEY further. Therefore, a mathematical definition of MEY
would be helpful in setting yield goals for specific costprice
situations. When maximum yield is reached an increase of
production inputs does not increase yield. Furthermore, the ratio
of change in output/change of input becomes less as maximum yield
is approached and is equal to zero at maximum yield.
It is not economical to add more production inputs when the
cost of added inputs is greater than the value of the increase in
output. Therefore, MEY occurs when the cost of added input is
equal to the value of the yield increase produced. This can be
expressed mathematically as follows:
(AY)(Yp) = (AX)(Xc)
where AY = yield increase
AX = increase of production input
Yp = commodity price
Xc = production cost per unit of input.
Making use of calculus the above equation can be expressed as dy/dx
= Xc/Yp
where dy/dx = slope of the yield versus input
function and
Xc/Yp = cost/price ratio.
A mathematical definition of MEY is illustrated for grain
yield of irrigated corn in figure 1 where the production input is
nitrogen (N). Maximum yield and MEY for a cost/price ratio of 0.58
are shown. Nitrogen rate and MEY for a cost/price ratio of 0.58
are 150 lb/A and 203 bu/A, respectively. The MEY for a cost/price
ratio of 1.0 is 120 bu/A and nitrogen rate is 45 lb/A. To
determine MEY for any cost/price ratio, construct a horizontal line
from the cost/price axis to the curve representing the cost/price
function, then construct a vertical line from the C/P curve to the
yield curve, finally construct a horizontal line from the yield
curve to the yield axis, the resulting value is the MEY. Nitrogen
rate is equal to the distance between the yield axis and the
vertical line connecting the two response curves.
Some input costs remain fixed regardless of yield goal. These
costs include equipment, land, pesticide, etc. Variable costs such
as nitrogen, phosphorus, potassium, seed, and irrigation change
with yield goal. Nitrogen is usually the factor that limits yield
of corn most. Therefore, when N rate is reduced other inputs can
be reduced in proportion to the N reduction. Consequently, all
variable input costs can be expressed in terms of cost per pound of
N.
Magnitude of profit margin, which may be positive or negative,
depends on the difference between total input costs and total
output returns. Remember, when profit is negative low or high
input agriculture is not sustainable. The effect of increasing N
cost on profit from irrigated corn at $2.00/bu is shown in table 1.
As expected, the cost/price ratio increased as the cost of N
Yield (bu/acre)
Cost/Price (ratio)
Maximum Yield 244 bu/A N 294 Ib/A
0 50 100 150 200 250 300
1.4
1.2
1
0.8
0.6
0.4
0.2
0
350
Ib N/acre
Figure 1. Grain yield of corn as a function of nitrogen rate. The cost/price
ratio is equal to the slope of the yield function. Equations were
derived from yield data obtained at the North Florida Research and
Education Center, Quincy.
300
increased, while profit and MEY decreased. Profit also decreased
as N rate decreased. Zero profit occurred with 120 lb of N/A and
a fixed cost of $200/A. However, a net loss of $58/A was the
result of zero N rate and $200/A fixed cost. The break even corn
price with zero N and $200/A fixed cost is $2.82/bu or the fixed
cost would have to be reduced to $142/A in order to break even with
zero N at $2.00/bu. Fixed costs can vary between locations due to
pest control and land cost differences. A higher N rate was
required to break even with a $300/A fixed cost than with a $200/A
fixed cost (198 vs. 120 lb/A). The influence of corn price on
profit and MEY of irrigated corn is shown in table 2. At $3.00/bu
MEY is equal to maximum yield which requires about 270 lb N/A.
However, further increases in corn price do not justify additional
production inputs because maximum yield has occurred and additional
inputs only decrease profit.
Table 1. Influence of N cost on MEY and net profit for corn
production at two levels of fixed input costs.
Calculations are based on functions shown in figure 1.
Corn price is $2.00 per bushel.
N Rate Cost of N Cost/price MEYt Fixed Input Costs
lb/acre $/lb ratio bu/acre $200/acre $300/acre
profit $/acre
261 0.25 0.13 242 218.75 118.75
232 0.50 0.25 237 158.00 58.00
199 0.75 0.38 226 102.75 2.75
169 1.00 0.50 213 57.00 43.00
107 1.50 0.75 174 12.50 112.50
45 2.00 1.00 120 50.00 159.00
tUnder certain cost structures MEY represents the minimum loss.
iProfit = (bu/acre x $/bu) (lb N/acre x $/lb N + other input
costs).
Table 2. Influence of corn price on MEY and net profit at two
levels of fixed input costs. Calculations are based on
functions shown in figure 1. Price of N is $0.30 per lb.
N Rate Corn price Cost/price MEYt Fixed Input Costs
lb/acre $/bu ratio bu/acre $200/acre $300/acre
profit $/acre
219 1.00 0.30 233 32.70 132.70
244 1.50 0.20 239 85.30 14.70
256 2.00 0.15 241 205.20 105.20
269 3.00 0.10 243 448.30 348.30
269 4.00 0.08 243 691.30 591.30
tUnder certain price structures MEY represents the minimum loss.
Profit = (bu/acre x $/bu) (lb N/acre x $/lb N + other input
costs).
This report shows conclusively that MEY is inversely proportional
to the cost/price ratio for production of irrigated corn.
Furthermore, in production systems adequately described by the
function in figure 1, it is not profitable to add variable inputs
when the cost/price ratio is greater than 1.18 regardless of fixed
costs. Data are needed to develop regression models of yield as a
function of pesticide rates in order to define low input
sustainable agriculture (LISA).
