2024年1月15日发(作者：)
肥料经济学（Ⅰ）--最大利润与最高产量
Murray Fulton教授
加拿大萨斯卡彻温大学农经系、合作研究中心
引言 Introduction
肥料对中国的农业和经济具有极为重要的意义。过去40年来中国农业增长的主要原因之一是肥料使用量的增加。目前中国的化肥年使用量约9千万吨（以产品重量计），成为世界上化肥消费的大国之一。同时，中国也是世界上进口化肥最多的国家，进口量约为1500万吨。
Fertilizer is extremely important to Chinese agriculture and to the Chinese economy. One of the major reasons
for the increase in agricultural production over the last 40 years is the increase in use of fertilizer. Currently
China consumes approximately 90 million metric tons (mmt) of chemical fertilizer (measured in production weight).
This makes China one of the largest consumers of fertilizer in the world. China is also the largest importer
of fertilizer, importing roughly 15 mmt of chemical fertilizer.
除了为作物生产提供养分外，化肥生产和运销在中国经济中也具有重要意义。例如，中国现有近1800个化肥厂，从业人员达220万，政府每年花费数十亿元用于肥料的生产和分销，化肥进口也是中国外汇硬通货支出的重要项目之一。
In addition to providing the nutrients required for crop production, fertilizer production and transportation
play an important role in the Chinese economy. The nearly 1,800 fertilizer plants in China, for instance, employ
2.2 million workers. Government investment into fertilizer production and distribution involves expenditures
of billions of yuan per year, while the importation of fertilizer is one of the major uses of hard foreign currency.
肥料生产经济学 The Economics of Fertilizer Production
经济学的主要特性之一是着重研究：为达到某一目标，如何最有效地分配和利用资源，不管是资金、劳动力或肥料。经济学的一个特点是可以用同样的概念和方法针对多种不同的问题或目标进行分析。
A key feature of economics is its focus on how best to allocate resources- whether this be money, or labour,
or fertilizer - in order to achieve some goal. One of the lessons of economics is that a large number of problems
- ie., many different goals - can be analysed with the same set of tools and concepts.
具体地说，经济学可对肥料使用中两个完全不同的问题做出解答。其一是应该使用多少肥料以获得最大量的物质指标，如单产量或总产量；其二是应该使用多少肥料以获得最大量的经济指标，如农户的利润，或者一个省或一个国家的纯经济效益。
In particular, economics can provide an answer to two very different questions that are often raised with
respect to fertilizer. The first concerns how much fertilizer should be used to maximize a physical variable
such as yield or total production. The second concerns how much fertilizer should be used to maximize an economic
variable such as the profits of the farmer or the economic net benefits of a province or a country.
要对这两个问题做出回答，通常的方法是从边际分析入手。就肥料的使用而言，边际分析的重要一步是要了解产量曲线或生产函数。如下面将要显示的，产量曲线的形状与边际收益递减的概念有着密切的关系。
The common approach to answering both of these questions involves the application of what is known as marginal
analysis. In the case of fertilizer, an important part of marginal analysis involves an understanding of the
shape of the yield response curve, or production function. As will be shown, the shape of the yield response
curve is closely related to the idea of diminishing marginal returns.
鉴于生产函数的重要性，这里首先对它进行分析。在对产量和肥料之间的物质关系了解之后，再对最高产量和最大利润进行阐述。同时，还将对产量优化和利润优化的不同解决方法进行讨论。
Because of its importance, the production function is examined first. Once the nature of the physical
relationship between fertilizer and yield is understood, the analysis shifts to an examination of the question
of maximizing yield, followed by an examination of profit maximization. An explanation of the difference between
the solutions to these two problems is also discussed.
在展开前面所提及的概念时，会用到中国农业的一些例子，这些资料大部分来源于加拿大钾磷研究所的研究结果，主要涉及钾肥的施用。
In developing the concepts outlined above, examples from Chinese agriculture will be used. Many of the examples
will be taken from work done by the Potash and Phosphate Institute of Canada (PPIC) and will involve the use
of potash.
这个讲座所介绍的概念带有理想化的色彩，也就是说，我们假设肥料的使用是没有限制的。下一个讲座，即“肥料经济学（Ⅱ）”，将讨论在存在限制因素的情况下怎样进行分析。
The concepts presented in this lecture will consider an ideal world in which there are no constraints to
the use of fertilizer. The next lecture, Economics of Fertilizer Ⅱ, will discuss how the analysis needs to be
changed in order to examine the presence of constraints.
在讨论合理施肥的经济问题时，我要强调这些讲座主要以概念论述为主。也就是说，我们将着重考虑肥料使用中的问题而不涉及细节。因此对于诸如价格或产量的不稳定性、施用时间、施肥位置和深度以及环境因素如有效土壤湿度的影响等等将不予讨论。这些问题都很重要，很多专著已作了研究，在此过多考虑它会分散对本讲座经济学思维框架主题的讨论。
In discussing the economics of fertilization, I should stress that the lectures will be largely conceptual
in nature. By this I mean they will focus on a way of thinking about problems regarding fertilizer use, rather
than providing a great number of specifics. As a result, I will not be discussing factors such as the impact
of price or yield uncertainty, the timing, depth and placement of fertilizer application, nor the effect of
environmental factors such as moisture availability. While these are important topics about which a great deal
has been written, consideration of them would detract from my main purpose of providing a framework for thinking
about problems in an economic fashion.
生产函数 The Production Function
生产函数也称产量反应曲线，在这里表示作物产量与肥料施用量之间的内在关系。它在以下讨论中非常有用，故应以数学形式表示，其表达式如下：y=f（x）
The production function is also known as the yield response curve. The production function shows the
relationship that exists between the yield of a crop and the amount of fertilizer used. Since it will be useful
later to be able to write this relationship in a mathematical form, the production function can be summarized
by the expression
y = f（x）
其中，y表示每亩的产量，x为肥料施用量，f（x）表示y是x的函数。
where y is the yield per acre and x is the amount of fertilizer that is applied. The expression f (x) means
that y is a function of x.
表1表示水稻产量与钾肥施用的关系，资料来源于中加钾肥项目1986年在湖北省的试验结果。图1是根据同样的资料绘制的。
Table 1 shows the relationship between rice yield and potash fertilizer usage in Hubei province in 1986.
This relationship was established through field research supported by the China/Canada Potash Agronomy Program.
Figure 1 graphs this same information.
(表：表1 钾肥对水稻的生产函数 （ 1986年 湖北） )
Fertilizer Use 施肥 Rice Yields 水稻产量 Yield Change 产量变化量 MPP边际产量
N P2O5 K2O Early早稻 Late晚稻 Early早稻 Late晚稻 Early早稻 Late晚稻
0
60
5151
5576
6105
6633
6942
-
425
283
-
528
309
7.08
4.72
8.80
5.15
kg/ha
150 60
150 60
150 60 120 5859
(图：图1 钾肥对水稻的生产函数（1986年，湖北）)
表1和图1中的数据所表现的一个重要特性是：随着肥料施用量的增加，追加的肥料所带来的水稻产量的增加量逐渐减小。例如，晚稻每公顷开始施用60公斤氧化钾（K2O）时，产量增加528公斤/公顷。继续增加60公斤/公顷时，产量只增加309公斤/公顷。
One of the most important features of the data in Table 1 and Figure 1 is that rice yields become less responsive
to the addition of fertilizer as fertilizer use increases. For instance, in late rice, the application of the
first 60 kg/ha K2O adds 528 kg/ha to the rice yield. The application of an additional 60 kg/ha, however, adds
only 309 kg/ha to the rice yield.
产量的增加量随着肥料施用的增加而逐渐减少的趋势称为边际报酬递减。使用边际物质产量（MPP）的定义将使这一概念的表述更加简明:
The tendency for the yield response to become less as fertilizer use increases is known as diminishing marginal
returns. The concept of diminishing marginal returns can be made more succinct by defining the marginal physical
product (MPP)
边际物质产量(MPP)=作物产量的变化量 / 肥料用量的变化量
Marginal Physical Product (MPP)=Change in Crop Yield / Change in Fertilizer Use
表1所示为湖北省早、晚稻钾肥施用的边际产量。计算得到一定肥料用量水平的边际产量。公式中的肥料用量变化量是指一定肥料用量水平与上一用量水平之差。作物产量变化量是指这两个肥料水平对应的两个产量的差值。
Table 1 shows the calculation of the MPP for potash on early and late rice in Hubei. The MPP is calculated
for a given level of fertilizer use. The change in fertilizer used in the calculation of MPP is the difference
between the given level of fertilizer use and the next smaller level of fertilizer use. The change in yield is
calculated as the difference between the yields at these two fertilizer usage levels.
例如，用309除以60而得的边际产量为5.15。其中，309是6942与6633之差，它们是钾肥用量分别为120和60公斤/公顷时的相应产量，60是120与60之差。
For example, the MPP of 5.15 is calculated by dividing 309 by 60. The value of 309 is the difference between
6942 and 6633, the yields at potash application levels of 120 and 60 kg/ha, respectively. The value of 60 is
the difference between 120 and 60.
应该强调的是，边际产量的计算并非是用一定的肥料用量水平下的产量与不用肥料的情况下的产量之差除以总的肥料用量。以表1为例来说，用837（即6942-6105）除以120（即120-0）而得出的6.98并不是边际产量。尽管这个数值在计算价值成本比（VCR）时很重要，但在肥料经济学中却并没有实际意义。下面我们会看到，用价值成本比来考察肥料经济会导致极不正确的结论。
It should be stressed that the calculation of the MPP does not involve dividing the difference between the
yield at a given level of fertilizer and the base yield when no fertilizer is applied by the total amount of
fertilizer used. In the example in Table 1, this latter number is obtained by dividing 837 (6942-6105) by 120
(120-0) to give 6.98. While this number plays an important role in the calculation of the Value Cost Ratio (VCR),
it plays no useful role in understanding the economics of fertilizer application. As will be shown below, the
VCR can produce extremely misleading results when it comes to considering the economics of fertilization.
当肥料用量水平差值较大时，通常考虑计算超出两个肥料用量水平范围的边际产量。例如，表1应该解释为，当晚稻的钾肥用量为0~60公斤/公顷时，边际产量为8.80；当钾肥用量为60~120公斤/公顷时，边际产量为5.15。
When the difference in the level of fertilizer use is large, it is usually assumed that the MPP that is
calculated applies over the range between the two levels of fertilizer use. For example, Table 1 should be
interpreted as saying that the MPP of late rice is 8.80 for fertilizer application levels between zero and 60
kg/ha. For fertilizer application levels between 60 and 120kg/ha, the MPP is 5.15.
边际产量的重要性在于它表示了肥料的追加用量所带来的产量的增量，如表1中所表现的逐渐下降的边际产量说明，肥料用量的增加所带来的增产幅度越来越小。简言之，边际产量表示，每增加一个单位的肥料用量在产量增加中的作用如何。
The importance of the MPP is that it shows what the incremental yield is from an additional application of
fertilizer. For instance, the falling MPP presented in Table 1 indicates that the increase in yield that results
from an increase in fertilizer usage is becoming smaller and smaller. In short, the MPP provides an indication
of how effective additional units of fertilizer are in raising yield.
从图形上看，生产函数曲线的斜率即为边际产量。图1中，线段AB的斜率是钾肥用量为0~60公斤/公顷时的边际产量，线段BC的斜率是钾肥用量范围为60~120公斤/公顷时的边际产量。图2绘出早、晚稻的钾肥边际产量与肥料水平的关系，这些曲线斜率的下降表明了边际报酬递减。
Graphically, the MPP is given by the slope of the production function. In Figure1, the slope of the line
AB gives the MPP of potash over the range of potash use zero to 60kg/ha, while the slope of the line BC gives
the MPP of potash over the range of potash use 60 to 120kg/ha. Figure 2 graphs the MPP of potash against the
level of fertilizer for both early and late rice. The downward slopes of these curves indicate diminishing marginal
returns.
(图：图2 钾肥对水稻的边际产量（1986年，湖北）)
图3表示的是钾肥的一般生产函数。用数学的术语来表示，边际产量是生产函数的一阶导数，即：
Figure 3 presents a more general production function for potash. In mathematical terms, the marginal physical
product is given by the first derivative of the production function, ie.,
(图：图3 钾肥对水稻的一般生产函数)
因此，线段AB的斜率为MPP1，是钾肥用量为K1时的边际产量；线段CD的斜率为MPP2，是钾肥用量为K2时的边际产量。钾肥用量为K3时，边际产量为负值，即MPP3＜0。再说明一下，边际产量与肥料用量水平的关系可以用图表示，边际报酬递减表明边际产量曲线应该向下倾斜。
Thus, the slope of the line AB is MPP1, the marginal physical product of fertilizer at a level of fertilizer
use K1, while the slope of the line CD is MPP2, the marginal physical product of fertilizer at usage level K2.
At fertilizer usage K3, the MPP becomes negative, ie., MPP3＜0. Once again, the MPP can be graphed against the
level of fertilizer usage. Diminishing marginal returns indicate that the MPP curve will be downward sloping.
图4可用来表示边际产量与用于计算价值成本比（VCR）的主要成份之间的关系。假定肥料的三个用量水平为K1、K2和K3，相应的产量分别为Y1、Y2和Y3。Y0为不施用肥料时的产量水平。计算价值成本比的第一步是计算下列比值： (Y1-Y0) / K1，(Y2-Y0) / K2，(Y3-Y0) / K3。这些比值分别为斜线Y0a、Y0b和Y0C。从图4中可以看到，这些线段的斜率和切线AB、CD及EF的斜率没有任何的关系。例如，肥料用量为K3时，边际产量为负值，而 (Y3-Y0)/K3却是正值。
Figure 4 can be used to show the relationship between the MPP and a major component used in the calculation
of VCR. Suppose that three levels of fertilizer use have been observed - K1, K2, and K3. Associated with these
levels of fertilizer use are yields Y1, Y2, and Y3, respectively. Yield level Y0 is the yield that results when
no fertilizer is applied. The first step in calculating the VCR is to calculate the value of the ratios: (Y1-Y0)/K1,
(Y2-Y0)/K2, and (Y3-Y0)/K3. The values of these ratios are given by the slopes of the lines Y0a, Y0b, and Y0c. As can
be seen from Figure 4, the slope of these lines bears no relation at all to the slope of the tangent lines AB,
CD, and EF. For example, at fertilizer usage level K3, the MPP has become negative. However, the ratio (Y3-Y0)/K3
is positive.
(图：图4 计算价值成本比所需要的要素)
多种肥料的生产函数 Production Function with Multiple Fertilizers
在研究肥料用量与作物产量之间的关系时，有必要明确有关的重要变量。例如图1和表1的数据表示两个生产函数，一个是早稻，一个是晚稻。另外，这两个生产函数都是在一定的氮磷肥用量水平下，按照某个特定的省（湖北）在某个特定年份（1986）的实验而得出。
When presenting the relationship between fertilizer use and crop yield, it is important that all the variables
important in the relationship be understood. For instance, Table 1 and Figure 1 present data for two production
functions, one for Early Rice and one for Late Rice. In addition, the production functions shown are based on
a given amount of nitrogen and phosphorus fertilizer and are based on trials in a particular province (Hubei)
in a particular year (1986).
生产函数可以通过显示产量和所研究的各种肥料不同用量水平之间的关系，来说明不同肥料之间的相互作用。例如，表2是施用氮和钾与黄麻产量的关系，资料来源于中加钾肥项目1986年在广东省的大田试验。
Production functions can account for the interaction between different types of fertilizer by showing the
relationship between yield and the level of usage of all the fertilizers under consideration. As an example,
Table 2 shows the relationship between the yield of jute and the amount of nitrogen and potash applied. The data
is for Guandong in 1986 and was established through field research supported by the China/Canada Agronomy Program.
(表：表2 黄麻产量作为N和K2O用量的函数（1986年，广东） )
K2O Level 施钾量 Yield (kg/ha) 产量 MPP of Potash 钾的边际产量 MPP of Nitrogen 氮的边际产量
(kg/ha)
0
75
150
225
N=150
3585
5160
6015
6135
N=225
3532
5182
6914
7107
N=150
21.0
11.4
1.6
N=225
22.0
23.1
2.6
-0.7
0.3
12.0
13.0
图5画的是多个生产要素的生产函数关系。为了便于理解，可以把图5看成是6个单生产要素的生产函数：2个为钾在2个不同的氮水平上，4个为氮在4个不同的钾水平上。线段ABCD和A'B'C'D'表示钾的两个生产函数，线段AA'、BB'、CC'和DD'表示氮的四个生产函数。
Figure 5 graphs this relationship in a multiple input production function. It is useful to think of Figure
5 in terms of six single input production functions: two for potash given the two different levels of nitrogen
application; and four for nitrogen given the four different levels of potash application. The lines ABCD and
A'B'C'D' show the two production functions for potash, while the lines AA', BB', CC', and DD' show the four
production functions for nitrogen.
(图：图5 氮肥和钾肥对黄麻的生产函数（1986年，广东）)
六个单生产要素的生产函数可计算六组边际产量，例如，在氮用量为150公斤/公顷时，可计算一组钾的边际产量，在氮用量为225公斤/公顷时，可计算另一组边际产量。表2提供了两组边际产量值。
The six single input production functions allow for the calculation of six sets of MPPs. For instance, one
set of MPPs can be calculated for potash, given that the application of nitrogen is 150kg/ha, while another set
of MPPs can be calculated for potash, given that the application of nitrogen is 225kg/ha. Table 2 presents the
two sets of MPPs.
当氮用量为150公斤/公顷时，黄麻产量对钾的用量在所有的钾用量水平均呈边际报酬递减。而在氮用量为225公斤/公顷时，黄麻产量起初对钾用量出现边际报酬递增。这是因为，边际产量在钾用量为75~150公斤/公顷时比钾用量为0~75公斤/公顷时大。然而，在钾用量为150~225公斤/公顷时，又出现边际报酬递减，在此钾肥施用范围，边际产量减至2.6。
For a nitrogen application of 150kg/ha, jute yields experience diminishing marginal returns to potash
application over all levels of potash application. For a nitrogen application of 225kg/ha, however, jute yields
initially experience increasing marginal returns to potash application. This follows because the MPP for potash
use between 75 and 150kg/ha is greater than the MPP for potash use between zero and 75kg/ha. For potash applications
between 150kg/ha and 225kg/ha, however, there are diminishing marginal returns to potash application. In this
range of fertilizer use, the MPP falls to 2.6.
图5中，当氮的用量为150公斤/公顷时，钾用量的边际报酬递减反映在生产函数ABCD的斜率随着钾肥用量的增加而逐渐变小。当氮用量为225公斤/公顷时，边际报酬的递增表现为生产函数的斜率从A'到B'，再到C'的逐渐增加，从C'到D'斜率的下降表明边际报酬开始递减。
In Figure 5, the diminishing marginal returns to potash application when nitrogen application is 150kg/ha
are reflected by the slope of the production function ABCD becoming smaller and smaller as the amount of potash
applied increases. For a nitrogen application level of 225kg/ha, increasing marginal returns are shown by the
increase in the slope of the production function as one moves from A' to B' to C'. The fall in the slope as one
moves to D' indicates that diminishing marginal returns have set in.
表2又表示在钾的各个用量水平，氮肥用量的变化对钾肥边际产量的影响。特别是氮肥用量的增加具有提高钾肥边际产
量的作用。例如，当钾肥用量为75~150公斤/公顷时，氮肥用量由150公斤/公顷增加到225公斤/公顷使得钾肥的边际产量由11.4增至23.1。
Table 2 also shows that a change in the level of nitrogen application affects the MPP of potash use for any
given level of potash use. In particular, an increase in the amount of nitrogen applied has the effect of increasing
the MPP of potash. As an example, when potash use is between 75kg/ha and 150kg/ha, an increase in nitrogen
application from 150 to 225kg/ha increases the MPP of potash from 11.4 to 23.1.
没有足够的数据来证明黄麻的产量是否对氮肥施用亦呈边际报酬递减。然而，表2表明，当不施钾时，氮肥的边际产量为负值，也就是说，氮用量由150公斤/公顷增加到225公斤/公顷导致了黄麻产量的降低，而随着钾用量的增加，氮肥的边际产量也增加。
Insufficient data is available to determine whether jute yields are also experiencing diminishing returns
to nitrogen application. Table 2 shows, however, that when no potash is applied, the marginal physical product
of nitrogen is negative -- that is, an increase in nitrogen use from 150kg/ha to 225kg/ha leads to a decrease
in jute yields. As the amount of potash applied increases, the MPP of nitrogen increases.
氮肥的边际产量随着钾肥用量的增加而增加的现象说明，钾肥对氮肥施用的有效性具有积极的作用。如前所述，氮肥对钾肥的有效性同样具有重要作用。这种内在关系的重要性在于，在决定某种肥料的用量时，必须同时考虑到其它肥料的用量。这种相互依存关系在后面还会作进一步的分析。
The increase in the MPP of nitrogen as potash application levels are increased is an indication of the positive
impact that potash has on the effectiveness of nitrogen. As was just discussed, nitrogen also has an important
effect on the effectiveness of potash. The importance of this interrelationship is that decisions about how much
of one fertilizer to use cannot be made independently of the decision regarding the amount of the other. This
interdependence will be further examined at a later point.
图6表示一个氮和钾两种肥料的一般生产函数。这个生产函数可用数学形式表示为
y=f(N，K)
Figure 6 presents a general production function for two fertilizers, potash and nitrogen. The production
function can be written mathematically as
y=f(N，K)
(图：图6 两种相互作物的肥料的一般生产函数)
氮肥的边际产量（MPPN）是这个函数在“氮方向”的斜率，因而，在氮肥用量为N时，MPPN等于线段CD的斜率。钾的边际产量（MPPK）是这个函数在“钾方向”的斜率，即MPPK等于线段AB的斜率。
The MPP of nitrogen (MPPN) is given by the slope of the function in the "N direction." Thus, at N*, MPPN is
equal to the slope of line CD. The MPP of potash (MPPK) is given by the slope of the function in the "K direction,"
ie., MPPK is equal to the slope of line AB.
数学上，这些边际产量是生产函数的偏导数，
Mathematically, the MPPs are given by the partial derivatives of the production function. Thus,
边际报酬递减是说对N和K的偏导数分别随着N和K的增加而减少。
Diminishing marginal returns implies that the partial derivatives with respect to N and K decrease as N and
K, respectively, increase.
最高产量 Output Maximization
如我在引言中提到的，边际报酬递减的概念是理解肥料经济学的关键。报酬递减表明，在一定程度上，肥料单位用量的增加不再对产量产生影响，即产量增量为零。如果产量不再增加，那一定是达到了最高产量。由于产量增加为零也意味着边际产量为零，达到最大产量时也就是肥料的边际产量为零的时候。图7表明，当肥料用量为K
max时，生产函数的斜率（线段EF的斜率）为零，即边际产量为零，因而，K
max便为获得最高产量的肥料用量。
As I mentioned in my introduction, the concept of diminishing marginal returns plays a key role in
understanding the economics of fertilization. Diminishing returns suggest that at some point an additional unit
of fertilizer no longer has any impact on yield -- i.e., the yield response is zero. If yield cannot be increased,
then the maximum yield must be obtained. Since a zero yield response also means the MPP is zero, then the maximum
yield is attained when the MPP of fertilizer is zero. Figure 7 shows that at fertilizer level K
max the slope
of the production function (the slope of the line EF) is zero and the MPP is zero. Fertilizer level K
max is therefore
the level of fertilizer usage that maximizes production.
*
(图：图7 钾肥对水稻的一般生产函数)
当两种或两种以上的肥料相互作用而影响产量时，寻找取得最高产量的肥料用量的方法也是相似的。在这种情况下，最高产量产生在两种肥料的边际产量都等于零的时候。因为，如果两个边际产量都为零，说明产量对任何一种肥料的施用都不再做出反应。
The method of finding the level of fertilizer use that maximizes production is similar when two or more
fertilizers interact to affect yield. In this situation, maximum yield occurs when the MPP for both fertilizers
are equal to zero. This is because if both MPPs are equal to zero, then yield no longer responds to the application
of either fertilizer.
图8表示的是产量为氮肥和钾肥施用的函数。当“钾肥方向”的线段斜率（即线段AB）和“氮肥方向”的线段斜率（即线段CD）同时为零时，取得最高产量，其结果是钾的用量为K
max和氮的用量为N
max时，产量最高。
Figure 8 presents a general production function that has yield as a function of nitrogen and potash use.
Maximum yield is attained when the slope of the line in the "potash direction" (i.e., line AB) is equal to zero
and when the slope of the line in the "nitrogen direction" (i.e., line CD) is equal to zero. As a result, an
application of K
max
of potash and N
max of nitrogen will maximize yield.
(图：图8 两种肥料的生产函数达到最高产量的肥料用量水平)
最大利润 Profit Maximization
经济学家们通常认为，在大多数情况下最高产量并不是一个适当的目标。而更相信投入值和产出值有某种关系。因为投
入的生产要素一般来说不是免费的，需要消耗资源，无论这种消耗是金钱、时间或放弃生产其它可供选择的产品，生产要素的价值即为成本。另一方面，产品是对资源的潜在索求，无论是以金钱或以其它产品的形式，产品的价值即为收益。
Economists usually suggest that maximum production is not an appropriate objective in most situations. Instead,
they believe that an objective that values inputs and outputs in some manner is preferred. Since inputs are
generally not free, but require the expenditure of resources, whether it be in terms of money, time, or the giving
up of other production that could have been produced, the value of inputs is considered to be a cost. Outputs,
on the other hand, represent a potential claim on resources, whether it be in terms of money or other commodities.
As a result, the value of the output produced is considered to be a benefit.
对农户来说，产品的价值与生产要素的成本之差即为利润。就肥料而言，每公顷的利润可表示为
利润=PY-rX
其中，P为产品的价格，Y为每公顷产量，r为肥料价格，X为肥料用量。由于产量是肥料用量水平的函数（产量与肥料用量的关系表现在生产函数中），利润与肥料用量水平亦呈函数关系。
For an individual farmer, the difference between the value of the output and the cost of the inputs is referred
to as profits. In the case of fertilizer, profits per hectare can be written as
Profits = py-rx
where p is the price of the output, y is the yield per hectare, r is the price of fertilizer, and x is the
amount of fertilizer that is applied. Since yield is a function of the level of fertilizer use (the relationship
between yield and fertilizer use is given by the production function), profits will be a function of the level
of fertilizer use.
可以把利润看成是收益与成本之差，即：
利润=收益-成本
其中，收益=PY，成本=rX。公式中，收益等于所生产的产品的销售收入，成本等于为了生产产品而用于生产要素的费用。注意，固定成本如利息支付或土地支出并不包括在成本里。虽说固定成本很重要，但实际上不能改变，因此在推导取得最大利润的肥料用量时，并不起什么作用。
It is useful to think of profits as the difference between benefits and costs, i.e.,
Profits = Benefits-Costs
where Benefits = p y and Costs = r x. In this formulation, Benefits is equal to the revenue obtained from
the sale of the product being produced, while Costs is equal to the money that must be spent on inputs to produce
the output. Note that fixed costs such as interest payments or land payments are not included in Costs. Although
fixed costs are important, the fact that they are fixed means that they cannot be altered. As a result, they
play no role in understanding the amount of fertilizer to apply to maximize profits.
这里我应该指出，虽然利润是估计农户收益和成本的一个方法，在考虑整个省或整个国家时，需要用其它方法。在这种
情况下，收益和成本之差称为纯社会受益。计算纯社会受益时，要考虑到肥料使用的所有受益（包括所创造的就业机会和消费者食物的增加）以及所有成本，例如造成的环境污染。由于这个讲座并不是侧重于纯社会受益，我仅在此强调要用分析利润优化同样的方法来分析这个问题。
I should point out here that while profits are a way of valuing the benefits and costs for an individual
farmer, other measures need to be used when an entire province or country is being considered. In such cases,
the difference between the benefits and costs is known as net social benefit. In calculating net social benefit,
consideration has to be given to all the benefits of using fertilizer (including the benefits from the employment
that is created and the benefits from consumers having additional food), as well as all of the costs including
the cost of the pollution that is generated. While I will not focus too much on net social benefit in this lecture,
I will stress that an analysis of this goal uses the same methodology as does an analysis of profit maximization.
图9表示的是一个假设的水稻生产利润与钾肥用量水平之间的关系。标注“收益”的曲线表示各个钾水平的总收益，由生产函数乘上产品价格而绘出。由于产量对肥料表现为边际报酬递减，收益曲线对肥料也是边际报酬递减。
Figure 9 illustrates a hypothetical relationship between the profits from growing rice and the level of potash
fertilizer that is applied. The curve labelled Benefits shows the total benefits associated with each level of
potash. This curve is constructed by scaling or multiplying the production function by the price of the output.
Since yield experiences diminishing marginal returns to fertilizer, the Benefit curve also shows diminishing
marginal returns to fertilizer.
(图：图9 利润与钾肥用量的关系)
标注“成本”的曲线表示各个钾用量水平的总成本，由肥料用量乘以肥料价格绘制而成。标注“利润”的曲线为收益曲线和成本曲线之差。
The curve labelled Costs shows the total costs associated with each level of fertilizer use. It is constructed
by multiplying the level of fertilizer use by the price of fertilizer. The curve labelled Profits is the difference
between Benefits and Costs.
因为收益曲线对肥料表现为边际报酬递减，利润曲线也是边际报酬递减。更具体地说，利润曲线的斜率为收益曲线的斜率与成本曲线的斜率之差，这种关系的存在是因为肥料用量的变化量所引起的利润变化，等于由该变化量而产生的收益变化减去成本变化。
Because the Benefit curve shows diminishing marginal returns to fertilizer, the Profits curve also shows
diminishing marginal returns. More specifically, the slope of the Profits curve is given by the difference between
the slope of the Benefits curve and the slope of the Costs curve. This relationship holds because the change
in profits for a change in fertilizer use is equal to the change in benefits for a change in fertilizer use minus
the change in costs for a change in fertilizer use.
收益曲线的斜率称为边际产值（MVP）。边际产值表示肥料用量的变化所引起的收益变化，它等于产品价格乘上边际产量，即：
MVP=
The slope of the Benefits curve is called the Marginal Value Product (MVP). The MVP shows the change in benefits
for a change in fertilizer use. It is equal to the price of the output times the MPP, i.e.,
MVP = p MPP
成本曲线的斜率为r，即肥料价格。这是因为，每增加一个单位的肥料用量，成本的增加为r。利润曲线的斜率为MVP-r。
The slope of the Costs curve is r, the price of the input. This follows because the addition of one more
unit of fertilizer increases costs by an amount r. The slope of the Profits curve is thus MVP-r.
当肥料用量为K时，利润达到最大。在这一点，利润对肥料用量变化的反应为零，即利润曲线的斜率为零：MVP-r=0。也就是说，在K*这一点，收益曲线的斜率等于成本曲线的斜率，即MVP==r。
Maximum profits occurs at fertilizer level K*. At this point, the response of profits to a change in the
application of fertilizer is zero, i.e., the slope of the Profits curve is zero?MVP-r=0. This means that at K*,
the slope of the Benefits curve is equal to the slope of the Costs curve - that is, MVP= p MPP = r.
这个等式表明，当增加一个单位的肥料用量所得到的收益恰好等于这个单位用量的成本时，利润达到最大。MVP=r可改写为常用的形式
MPP=r/P
This expression says that profits will be maximized when the benefit from an additional unit of fertilizer
is just equal to the cost of that unit. It will be useful to rewrite the expression MVP = r as
MPP = r/p
表1的数据可用来示范怎样运用这一结果。假设水稻价格为0.5元/公斤，钾肥（K2O）的价格为0.73元/公斤，那么钾肥与水稻的价格比便为0.73/0.50=1.46。
The data presented in Table 1 can be used to show how this result can be applied. Assume that the price of
rice is 0.5yuan/kg, while the price of potash (K2O) is 0.73 yuan/kg. This means that the ratio of the price of
potash to the price of rice is 0.73/0.50=1.46.
如表1所显示，这个比值比早晚稻的边际产量都低，这表明，要取得最大利润，钾肥用量应超过120公斤/公顷。对图2作进一步地分析可得知，钾肥用量在120~180公斤/公顷之间时可能会产生最大利润。这一结论的得出是由于只有在钾肥*
用量超过120公斤/公顷时，才能使边际产量下降，从而与1.46相等。
As Table 1 indicates, this ratio is less than the MPPs that have been calculated for both late and early
rice. The implication is that potash application rates would have to be larger than 120kg/ha in order to obtain
maximum profits from the use of potash fertilizer. More specifically, an examination of Figure 2 suggests that
a potash application level of between 120 and 180kg/ha would likely result in maximum profits. This conclusion
is reached by noting that fertilizer application levels would have to increase beyond 120 kg/ha in order to reduce
the MPP to the point where it was equal to 1.46.
MPP=r/P的表达式使我们可以比较最大利润与最高产量。如图9所示，取得最大利润的肥料用量(K*)总是低于取得最高产量的肥料用量(K最大)。为了使产量达到最高，边际产量应该为零，而为了获得最大利润，边际产量应该等于肥料和产品的价格比。由于肥料的使用存在着边际报酬递减，增加肥料边际产量的唯一办法是少用肥料，因此，取得最大利润的肥料用量低于取得最高产量的肥料用量。
The expression MPP = r / p allows us to compare profit maximization with output or yield maximization. As
Figure 9 illustrates, the level of fertilizer use that maximizes profits (K*) is always less than the level that
maximizes output (K max). In order to maximize yield, the MPP should be set equal to zero. However, to maximize
profit, the MPP should be set equal to the fertilizer price/output price ratio. Since there are diminishing
marginal returns to the application of fertilizer, the only way to increase the MPP of fertilizer is to use less.
As a result, the profit maximizing level of fertilizer use is less than the output maximizing level.
MVP=r这个关系是从只使用一种肥料的情况推导出的，但也适用于两种或多种肥料相互作用而影响产量的情况。当使用多种肥料时，需要明确每种肥料的价格和边际产量。例如，如果有两种肥料，肥料价格可分别表示为r1和r2，边际产量可分别表示为MPP1和MPP2，边际价值产品分别为MVP1（等于P?MPP1）和MVP2（等于2）。
While the relationship, MVP = r, was derived for the case of a single fertilizer, it also holds for the case
where two or more fertilizers interact to affect yield. When there is more than one fertilizer, it is necessary
to specify the prices and the marginal physical products for each of the fertilizers. For instance, if there
are two fertilizers, the prices of these fertilizers could be specified as r1 and r2, while the marginal physical
products could be specified as MPP1 and MPP2. The marginal value products are MVP1 (equals p MPP1) and MVP2 (equals
p MPP2).
为了获得最大利润，两种肥料的用量都必须是在MVP1=r1和MVP2=r2的时候，换言之，
MPP1=r1/P, MPP2=r2/P
由于一种肥料的用量水平会影响到另一种肥料的边际产量，需要调节两种肥料的用量水平直到满足以上两个等式。
To maximize profits, it is necessary to find the level of both fertilizers where MVP1=r1 and MVP2=r2, or
alternatively, where
MPP1=r1/p and MPP2=r2/p
Since the level of one fertilizer affects the MPP of the other, it is necessary to alter the levels of both
fertilizers until the above equality is obtained.
图10中，获得最大利润的氮和钾用量分别为N*和K*。这个用量是由MPPN=rN/P和MPPK=rK/P而得出的。请注意到N*和K*都低于最高产量的用量水平N最大和K最大。
Figure 10 shows that the profit maximizing levels of nitrogen and potash are N* and K*. These levels are found
be equating rN / p and rK / p with MPPN
and MPPK, respectively. Note that both N and K* are less than N
max and
K
max, the levels of nitrogen and potash that maximize yield.
(图：图10 两种肥料的生产函数达到最大利润的肥料用量水平)
*
以上结果表明，只有在肥料价格为零时，最大利润才会和最高产量相一致，换言之，只有在肥料价格为零时，最高产量才有经济学上的意义。
The implication of the above result is that unless the price of fertilizer is zero, the maximization of output
is not consistent with maximizing profits. Or, put somewhat differently, maximizing output only makes sense from
an economic perspective when the price of fertilizer is zero.
经验证据 Empirical Evidence
产量优化与利润优化的问题很重要，因为我所参与的一项研究表明，至少在中国的某些地区，农户仍倾向于把氮肥用到最高产量的水平。由于肥料价格通常并不为零，最高产量意味着氮肥的使用已过量。这表明，可用于其它生产投入如钾或用于购买农产品或家用产品的资金被用在了氮肥上，其结果是农户所获得的利润减少。
The question of output versus profit maximization is important since research that I have been involved with
shows that farmers in at least some parts of China have tended to use nitrogen fertilizer to the point where
output is maximized. Since the price of fertilizer is usually not zero, the maximization of output means that
too much nitrogen is being used. The use of too much nitrogen means that resources (e.g., money) are being spent
on nitrogen that could be spent on other inputs such as potash, or used in the purchase of other farm or household
products. The result is a reduction in the level of profits that could be earned by farmers.
举例而言，表3是1988年湖南省中东部地区54个农户的施肥情况。其中，10户位于高产地区，早稻平均亩产达1045斤/亩。10户中9户用氮肥，5户用磷肥，6户用钾肥，9户用粪肥。
To give an example of fertilizer use in China, Table 3 presents data collected from 54 farms in east-central
Hunan province regarding their operations in 1988. Ten of the farms are located in a high yield region, where
the average yield of early rice is 1,045 jin/mu. Of the 10 farms in this area, 9 applied nitrogen, 5 applied
phosphate, 6 applied potash, and 9 used manure.
(表：表3 早稻的产量与肥料的用量（1988年，湖南省中东部地区） )
# Fertilizer 施肥户Stat. Diff. From Zero 与0的统计Range 产量幅度 Average 平均值 MPP 边际产量
数 学差异
(jin/mu)
High Yield Region (10 observations) 高产区（10个观察值）
Yield产量 900-1200
Nitrogen氮 0~57
Phosphate磷 0~50
Potash钾 0~20
1045.0
35.6
22.5
8.7
2000.0
-
9
5
6
9
-
-0.19
0.22
0.82
0.004
-
No无
No无
No无
No无 Manure有机肥 0~4000
Low Yield Region (44 observations) 低产区（44个观察值）
Yield产量 500~1000
Nitrogen氮 15~62
Phosphate磷 0~100
Potash钾 0~30
709.1
35.7
43.1
6.2
2657.0
-
44
32
19
40
-
2.44
0.01
1.26
0.001
-
No无
No无
YES有
No无 Manure有机肥 0-10000
注释：N是尿素，P是过磷酸钙，K是氯化钾。
在低产区的早稻亩产为500~1000斤/亩，虽然44户均施用氮，只有19户施钾，32户施磷。平均用氮水平为35.7斤/亩，与高产地区相接近。平均磷用量为43.1斤/亩，明显地高于高产地区的22.5斤/亩。然而，在施用磷肥的农户中，差别却小得多：高产区为45.0斤/亩，低产区为59.3斤/亩。
In the low yield region, early rice yields ranged from 500 to 1,000 jin/mu. While all 44 farmers in the region
applied nitrogen, only 19 applied potash. Thirty-two farmers applied phosphate. The average level of nitrogen
applied was 35.7 jin/mu, about the same as in the high yield region. The average level of phosphate application
was 43.1 jin/mu, considerably higher than the 22.5 jin/mu level in the high yield region. However, among those
farmers that used phosphate, the difference was much less: 45.0 jin/mu in the high yield region as compared to
59.3 jin/mu in the low yield region.
在高产区，施用钾肥的农户平均用量为14.5斤/亩，与低产区的14.4斤/亩接近。
In the high yield region, the average potash use among farmers using potash was 14.5 jin/mu, almost the same
as the 14.4 jin/mu average among the farmers that used potash in the low yield region.
表3还显示了以上两个地区四种肥料的边际产量。在高产区，四种肥料的边际产量都很小，统计上基本上是等于零，这说明，该地区的肥料用量与获得最大利润时所需要的用量相比已过量。
Table 3 also presents the MPPs that were estimated for the four different fertilizers in each of the two
regions. In the high yield region, the MPPs of all the fertilizers are small and cannot be differentiated from
zero statistically. The implication is that fertilizer is generally being overused in this region compared to
the level that would maximize profits.
在低产地区，磷肥和粪肥的边际产量都很小，统计上接近于零，说明这两种肥料的用量接近于最高产量的用量水平。由
于粪肥的价格可以看成是很低，这种养分的使用达到边际产量为零的水平并不奇怪；然而，磷肥的价格并不为零，磷肥的使用显然已经过量。
In the low yield region, the MPPs of phosphate and manure are small and cannot be differentiated from zero
statistically. This implies that these two fertilizers are being used at a level that would maximize yield. Since
the price of manure can be considered to be very low, it is not surprising that this nutrient source is being
used to the point where the MPP equals zero. However, given that the price of phosphate is not zero, it would
appear that this fertilizer is being overused.
对氮肥边际产量的测算相对较高，但由于数据的变异性，这个测算并不十分准确。因此可以排除氮肥的边际产量接近于零的可能，但不能排除其等于氮肥与水稻的价格比的可能。
The estimate of the MPP of nitrogen is relatively large. However, because of variability in the data, this
estimate is not very precise. As a result, the possibility that the MPP is zero cannot be rejected. Neither,
however, can the possibility that the MPP is equal to the nitrogen/rice price ratio be rejected.
对钾肥边际产量的估计是正值，而且统计上显然不等于零。这表明在追求最高产量时钾肥的使用并未过量。由于大部分农户并不用钾肥，为了了解农户用钾是否符合最大利润用量，对此进行了统计检验。检验结果表明确实如此。因此，对于那些未使用钾肥的农户，使用钾肥会增加他们的利润，也就是说，生产函数估计表明，不用钾肥不能取得最大利润。
The estimate of the MPP of potash is positive and statistically different from zero. This implies that potash
was generally not being overused in an attempt to maximize production. Since a large number of farmers did not
use potash, statistical tests were carried out to see if the farmers that used potash did so in a manner that
was consistent with profit maximization. It was found that this was the case. For the farmers that did not use
potash, however, access to potash fertilizer would have increased their profits, ie., the production function
estimates indicate that zero potash use does not maximize profits.
然而，如大家所知，农户能购买和使用的肥料量常常受到限制。在存在限制因素的情况下，合理施肥的经济问题也发生了变化。下一讲座的题目就是在存在限制条件的情况下怎样进行分析。
Unfortunately, as you know, farmers are often constrained in the amount of fertilizer that they can purchase
or use. When constraints are in place, the economics of fertilization changes. The subject of my next lecture
will be how to analyze situations in which constraints are present.
