肥料经济学肥料经济学Ⅱ存在限制因素情况下的利润优化

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1、肥料经济学（）-存在限制因素状况下的利润优化Murray Fulton专家加拿大萨斯卡彻温大学农经系、合伙研究中心引言 Introduction在肥料经济学的第一讲中，我论述了边际报酬递减的概念及其与生产函数形状之间的关系。具体地说，我阐明了边际报酬递减就是边际产量随着肥料用量的增长而递减。In my first lecture on the economics of fertilization, I examined the concept of diminishing marginal returns and how this is related to the shape of the

2、production function. More specifically, I showed that diminishing marginal returns is equivalent to a marginal physical product (MPP) that declines with increases in the level of fertilizer that is used.在肥料使用不受限制时，边际产量为零的肥料用量水平可以获得最高产量。然而，最高产量一般不是合适的目的。从经济学的观点来看，最高产量不能算计使用肥料的成本和收益，权衡成本和收益的常用措施是利润。 I

3、n a situation where no constraints on the use of fertilizer are present, production can be maximized by finding the level of fertilizer use that makes the MPP equal to zero. Maximizing production, however, is usually not the preferred goal. From an economic point of view, maximizing production fails

4、 to account for the costs and benefits of using fertilizer. The usual measure of the trade-off between these costs and benefits is profit.为了使利润值最大，某种特定肥料的边际产量必须等于这种肥料与产品的价格比，当这个比例拟定期，肥料的使用量要低于获得最高产量的肥料用量。In order to maximize profit, the MPP of a particular fertilizer should be set equal to the ferti

5、lizer price/output price ratio. When this is done, the amount of fertilizer used is less than that used to maximize production.边际产量应当等于肥料与产品价格比的论点是有假设条件的，即假设农户不受肥料量和资金的限制。正如上讲中讨论的那样，现状一般并非如此。世界各地的农民都面临着资金限制或买不到肥料。例如，在加拿大，部分农民受到贷款量的限制而不能随意购买化肥；在中国，由于缺少信贷，农民同样受到资金局限性的限制，此外，还也许受到化肥生产量或进口量局限性的限制。The arg

6、ument that the MPP should be set equal to the fertilizer price/output price ratio is based on the assumption that there are no constraints to the amount of fertilizer or the amount of money available to the farmer. As was discussed in the last lecture, this is usually not the case. Farmers all over

7、the world face limits on the amount of money or fertilizer to which they have access. In Canada, for instance, some farmers are restricted in the amount of credit they have available to purchase fertilizer. In China, farmers may also be constrained because of a lack of credit. They may also be const

8、rained because of a lack of fertilizer production or importation.在这一讲，我将分析在存在限制因素的状况下，如何使用肥料以达到最大利润。一方面讨论如何将有限的肥料在多种作物之间进行分派，这种分析还将延伸到用于肥料的资金如何在不同肥料之间进行分派，不管肥料是用于一种作物或是用于多种作物。In this lecture I examine the problem of how fertilizer should be used to maximize profits when constraints are in place. Fir

9、st, I examine the problem of how a fixed amount of fertilizer should be allocated among various crops. The analysis is then expanded to consider the allocation of fertilizer expenditures among different fertilizers, whether the fertilizer is applied to a single crop or to a number of crops.我们以简介基本概念

10、作为讨论这一主题的开始，以使这个问题易于理解。其中最重要的是机会成本概念。一方面，我要给机会成本下个定义，然后再用这一概念来分析如何在不同作物之间对肥料进行分派，以及如何在不同肥料之间对资金进行分派。I will begin my discussion of this topic by presenting a conceptual framework in which the problem can more easily be understood. The most important element of this framework is the notion of opportun

11、ity cost. After providing a definition of opportunity cost, I use this concept to examine how fertilizer should be allocated among different crops, and how fertilizer expenditures should be allocated among different fertilizers.有了以上概念后，再用湖南省中东部地区的资料来阐明这些概念是如何应用的。After I have outlined the conceptual

12、model, I will then use data on fertilizer use from a region in east-central Hunan province to illustrate how these concepts can be applied. 机会成本 Opportunity Cost经济学中最重要的概念之一是机会成本。下面会看到，这一概念是回答我所提出的问题的核心。One of the most important concepts in economics is opportunity cost. As I will show, this concept

13、 is key to answering the questions I have just posed.给机会成本下定义最佳从问题入手。水稻使用氮肥的成本是什么？答案显然是氮肥的价格，但这并不是真正的成本。真正的成本，即机会成本，是如果把这些氮肥用于另一种作物，如小麦，所能得到的额外产量。换句话说，在水稻上使用氮肥的真正成本是丧失了氮肥用于其他作物的机会。Opportunity cost is best defined with the help of a question. What is the cost of using nitrogen fertilizer on a rice cr

14、op? While the obvious answer is the price of nitrogen, this in fact is not the real cost. The true cost - or the opportunity cost - is the value of the extra output that could be produced if the nitrogen were used on another crop such as wheat. In other words, the real cost of using nitrogen on rice

15、 is the loss of the opportunity of using it on a different crop.机会成本但是是资源短缺的另一种说法。在上述例子中，只有在氮肥量有限时，把氮肥用于水稻才存在着机会成本。如果氮肥的供应没有限制，也就没有机会成本，由于这时氮肥用于水稻并不波及到放弃用于小麦的机会。Opportunity cost is simply another way of saying that resources are scarce. In the example above, using nitrogen on rice will only have an

16、opportunity cost if the amount of nitrogen is limited. If there was an unlimited supply of nitrogen, then there would be no opportunity cost, since using nitrogen on rice would not involve giving up the opportunity to use it on wheat.限制因素的存在阐明了资源的短缺。例如，如果农民的贷款量有限，用于购买诸如肥料等生产要素的资金就会短缺。如果政府能买回来分派的肥料量有

17、限，肥料就会局限性。The presence of constraints means resources are scarce. For instance, if farmers have limited credit, then money is scarce for the purchase of inputs such as fertilizer. If a government is constrained in the amount of fertilizer it is allocated or can purchase, then fertilizer is scarce.因此

18、，限制因素波及到机会成本。拟定这个机会成本是拟定如何分派有限资源以达到最大利润的第一步。Constraints, therefore, involve opportunity costs. Identifying this opportunity cost is the first step in deciding how limited resources should be allocated in order to maximize profits.可运用肥料的限制因素理论分析 Constraints in the Availability of Fertilizer - Theory肥

19、料限制最简朴的一种情形是生产一种作物，并且这种作物只需要一种肥料。在这种状况下，决策原则很简朴：如果可供运用的肥料量不不小于使利润达到最大值所需要的肥料量，则使用所有肥料。The simplest case of a fertilizer constraint is where one crop is produced and where this crop responds to only one kind of fertilizer. In this case the decision rule is straightforward: If the amount of fertilizer

20、 available is less than the amount that would maximize profits, then use all the fertilizer.(图：图1 肥料量有限状况下的利润优化)图1阐明了这种状况，达到最大利润的肥料水平是K*。如果可供使用的肥料量K constraint限制值不不小于K*，那么农民应当用K constraint。固然，如果K constraint限制值不小于K*，那么只该用K*，余下的部分应贮存到来年，或转售她人。Figure 1 illustrates this case. The level of fertilizer tha

21、t maximizes profits is K*. If the amount of fertilizer available, K constraint, is less than K*, then the farmer should use the amount K constraint. Of course, if K constraint is greater than K*, then only K* should be used. Any remainder should be stored for another year, or sold to someone else.图1

22、还表白，如果能以某种方式消除限制因素，就会增长收益。例如，如果可供使用的肥料增长一种单位，利润便会增长。利润的增长量为利润曲线在K constraint限制值这一点的斜率（即直线AB的斜率）。Figure 1 also indicates the benefit that would be received if the constraint could be removed in some way. If an additional unit of fertilizer were available, for instance, then profits would increase. Th

23、e amount of this increase in profits is shown by the slope of the Profits curve at K constraint (i.e., the slope of the line AB).如第一讲中所讨论的，利润曲线的斜率为收益曲线和成本曲线的斜率之差，即MVP-r。MVP和r都表达在图1的下图中。As was discussed in Economics of Fertilizer I, the slope of the Profits curve is given by the difference between th

24、e slope of the Benefits curve and the slope of the Costs curve - i.e., MVP - r. Both MVP and r are graphed in the lower portion of Figure 1.这里应当阐明一下，本讲座随后将用P表达产品价格，r代表肥料价格。如果作物和肥料不只一种，则相应用P1、P2和r1、r2来表达，以此类推。 I should point out here that in the remainder of this lecture, p will be used to refer to t

25、he price of the output, while r refers to the price of fertilizer. If more than one crop or fertilizer is discussed, the prices shall be denoted by p1, p2, r1, r2, and so on.由于在肥料用量为K constraint限制值时，MVP不小于r，因此，如果能以价格r多购买一种单位的肥料，利润便会增长。这是由于，肥料单位用量的增长使产量增长了MPP。这个增长产量的价值为MVP（MVPP.MPP），获得这个边际收益的成本为r，即肥料

26、的价格。由于MVP不小于r，因而收益不小于成本，利润增长，利润的增长量为MVP-r。Since MVP exceeds r at fertilizer level K constraint, profits would increase if an additional unit of fertilizer could be purchased at price r. This follows because an additional unit of fertilizer increases output by the MPP. The value of this additional ou

27、tput is given by MVP (which equals p MPP). The cost of obtaining this additional revenue is given by r, the price of fertilizer. Since MVP exceeds r, the benefit exceeds the costs and profits rise. The size of this increase in profits is MVP - r.明确额外肥料的收益有助于理解更复杂的问题。例如，假定目前有两种作物，两种作物都对同一种肥料有反映，如果肥料量

28、有限，应当如何在这两种作物之间分派这种肥料呢？Identifying the benefits of additional fertilizer is useful in understanding more complex problems. For instance, suppose now that two crops are produced and that both crops respond to the same fertilizer. If there is a limited amount of fertilizer available, how should this f

29、ertilizer be allocated between the two crops?这个问题的核心是想想有限的肥料量最初在两种作物之间的分派，这表达在图2中。图中的矩形表达固定的肥料量，虚线表达肥料在两种作物之间的分派。目前考虑把一公斤肥料从一种作物转移到另一种作物的成果。The key to this problem is to think about having a fixed amount of fertilizer that is initially allocated between two crops. Figure 2 illustrates this graphical

30、ly. The rectangle indicates a fixed amount of fertilizer, with the dotted line indicating the allocation of the fertilizer between the two crops. Now consider the impact of shifting one kilogram of fertilizer from one crop to the other.(图：图2肥料从一种作物转用到另一种作物的经济影响)假定第一种作物的边际产量为MPP1，第二种作物的边际产量为MPP2。把一公斤

31、肥料从作物1转到作物2，使作物1的产量下降MPP1，而作物2的产量增长MPP2。从产值上看，作物1的利润减少MVP1=P1.MPP1，作物2的利润增长MVP2=P2.MPP2。Suppose that the marginal physical product of the first crop is identified as MPP1, while the marginal physical product of the second crop is MPP2. Shifting one kilogram of fertilizer from crop 1 to crop 2 reduce

32、s the yield of crop 1 by MPP1 and increases the yield of crop 2 by MPP2. In value terms, the shift of fertilizer from crop 1 reduces profits by MVP1 (equals p1MPP1), while the shift of fertilizer to crop 2 increases profits by MVP2 (equals p2MPP2).只要MVP2不小于MVP1，就该继续把肥料从第一种作物移向第二种作物，但由于边际报酬递减，随着肥料转移的

33、增长，MVP2和MVP1之间的差别将变小。由于用于第二种作物肥料的增长使MPP2下降，而第一种作物肥料用量的减少使MPP1上升。As long as MVP2 is greater than MVP1, it will make sense to continue to shift fertilizer from crop 1 to crop 2. Because of diminishing marginal returns, however, the difference between MVP2 and MVP1 becomes smaller as more and more fert

34、ilizer is shifted. This occurs because the addition of fertilizer to crop 2 causes MPP2 to fall, while the removal of fertilizer from crop 1 causes MPP1 to rise. 为了使利润值最大，肥料应在两种作物间进行调节，直到MVP1=MVP2。如果两者不相等，利润总能通过肥料从一种作物移向另一种作物而得到增长。To maximize profits, fertilizer should be shifted between the two cro

35、ps until MVP1=MVP2. If these two expressions are not equal, profits can always be increased by moving fertilizer from one crop to the other.用机会成本的概念来考虑这个问题也很有用。肥料的机会成本是由于没有机会把肥料用于其他作物而损失的经济效益。在上例中，把肥料用于第一种作物的机会成本是放弃用于第二种作物应产生的收益MVP2，但把肥料用于第一种作物所提供的收益等于MVP1，当边际收益等于边际成本，即MVP1=MVP2时，利润达到最大。It is useful

36、 to consider this problem in terms of opportunity cost. Recall that the opportunity cost of fertilizer is the benefit foregone by not using the fertilizer somewhere else. In the above example, the opportunity cost of using fertilizer on crop 1 is the benefit given up from using fertilizer on crop 2,

37、 namely MVP2. Using fertilizer on crop 1, however, provides a benefit equal to MVP1. Profits will be maximized when the marginal benefit equals the marginal cost, i.e., MVP1=MVP2.图3用图示表达了这个问题。第一幅图表达第一种作物各个边际产值水平所需要的肥料量，第二幅图表达在第二种作物上同样的状况，第三幅图表达在第一、二种作物上产生等量边际产值所需要的肥料总量。第三幅图中的曲线是通过第一、二图中曲线的水平相加而得出的。

38、Figure 3 illustrates the solution to this problem in graphical form The first panel shows the amount of fertilizer required to generate each level of marginal value product for crop 1. The second panel shows the same thing for crop 2. The third panel shows the total amount of fertilizer required to

39、generate the same level of marginal value product in both crop 1 and crop 2. The curve in the third panel is drawn by horizontally adding together the curves in the first two panels.(图：图3 有限的肥料在两种作物之间的分派)假定可用的钾肥总量为Kconstraint，其中K1constraint限制量应当用于第一种作物，K2constraint应当用于第二种作物，由于K1constraint+K2constrai

40、nt=Kconstraint，所有的肥料均得到使用。此外，由于钾肥是在MVP1=MVP2的状况下进行分派的，任何肥料的重新分派都将导致利润下降。Suppose that the total amount of potash available is Kconstraint. Of this, an amount K1constraint should be allocated to crop 1, while an amount K2constraint should be allocated to crop 2. Since K1constraint+K2constraint=Kconstr

41、aint, all of the fertilizer is used. In addition, since MVP1=MVP2 with this distribution of potash any reallocation of fertilizer between the two crops will result in reduced profits.机会成本的概念可以用来解决其他有限制的问题。例如，假定某种作物（如水稻）需要两种肥料（氮和钾），并且这两种肥料互相作用以决定产量，再假定购买化肥的资金是有限的，那么，资金该如何在两种肥料之间进行分派呢？The notion of op

42、portunity cost can be used to solve other constraint problems. For instance, consider a crop (e.g, rice) that responds to two fertilizers (nitrogen and potash) which interact in determining yield. Suppose also that the amount of yuan available to spend on fertilizers is limited. How should the yuan

43、be spent between the two fertilizers?假定使用在N肥上的资金比使用在K肥上的多一元，如果N肥的价格为rN，有了这额外的一元钱，农民可以多买1/rN公斤的N肥，从而使水稻产量增长MPPN/rN公斤，这是多花一元在N肥上所产生的边际收益。Suppose an additional yuan is spent on nitrogen rather than on potash. If the price of nitrogen is rN, then with one more yuan, the farmer is able to purchase 1/rN m

44、ore kilograms of nitrogen. This, in turn, adds an additional MPPN/rN kilograms to rice output. This is the marginal benefit of spending an additional yuan on nitrogen.由于这一元钱被用于N肥，1/rK的K肥就不能买了（K肥的价格为rK），其成果是水稻产量下降MPPK/rK公斤，这就是机会成本。As a result of spending this yuan on nitrogen, 1/rK kilograms of potas

45、h can no longer be purchased (rK is the price of potash). The result is that rice output falls by MPPK/rK kilograms. This, of course, is the opportunity cost.机会成本等于边际收益便是最大利润的体现式，即当利润最大时MPPN/rN=MPPK/rK这个体现式也可改写为：MPPN/MPPK=rN/rK Equating the opportunity cost to the marginal benefit gives the expressi

46、on for maximizing profits. That is, profits will be maximized whenMPPN/rN=MPPK/rKwhich is equivalent toMPPN/MPPK=rN/rK表1用一种假想的例子阐明了这一原则。表的上半部分表达在不同N、K肥水平下N和K的边际产量。N和K肥均体现出边际报酬递减，N肥的边际产量随着N肥的增长而下降，K肥的边际产量也随着K肥的增长而减少。表的下半部分表达N肥边际产量（MPPN）和K肥边际产量（MPPK）之间的比例。Table 1 illustrates how this rule can be appli

47、ed using a hypothetical example. The entries in the top portion of Table 1 indicate the marginal physical products for nitrogen and potash at different levels of nitrogen and potash. Both nitrogen and potash exhibit diminishing marginal returns. The MPPN declines as nitrogen levels increase, while t

48、he MPPK declines as potash levels increase. The entries in the bottom portion of Table 1 represent the ratio of MPPN to the MPPK.(表：表 1不同肥料用量水平的边际产量（假设的例子） )N (kg/ha) 施氮量 氮的边际产量 MPPN 钾的边际产量 MPPK 氮钾边际产量比 MPPN/MPPK K2O (kg/ha) K2O (kg/ha) K2O (kg/ha) 75 150 225 75 150 225 75 150 225 75 1.94 1.98 2.01

49、1.06 0.69 0.31 1.8 2.9 6.4 150 1.64 1.68 1.71 1.10 0.73 0.35 1.5 2.3 4.9 225 1.34 1.38 1.41 1.14 0.76 0.39 1.2 1.8 3.6 300 1.04 1.08 1.11 1.18 0.80 0.43 0.9 1.3 2.6 375 0.74 0.78 0.81 1.21 0.84 0.46 0.6 0.9 1.8 假定如表2所示，N肥和K肥的价格比大概为1.78，再假定只有150元/公顷的有限资金用于肥料，那么，最合适的是购买大至为75公斤/公顷的N肥和75公斤/公顷的K肥，这是这笔有限资

50、金能产生的最大的利润。Suppose that nitrogen price/potash price ratio is approximately 1.78 as presented in Table 2. To see how this ratio is used, suppose there was only enough capital to spend 150 yuan/ha on fertilizer. In this case, the optimal purchase would involve approximately 75 kg/ha of nitrogen and 75

51、 kg/ha of potash. This would generate the maximum possible profits given the constraint on fertilizer expenditures.(表：表2 资金在两种不同肥料之间的分派 )N (kg/ha) 施氮量 氮钾边际产量比 MPPN/MPPK 耗费 Expenditures K2O (kg/ha) K2O (kg/ha) 75 150 225 75 150 225 75 1.8 2.9 6.4 152 207 262 150 1.5 2.3 4.9 250 305 359 225 1.2 1.8 3.

52、6 347 402 457 300 0.9 1.3 2.6 445 500 554 375 0.6 0.9 1.8 542 597 652 注释：Fertilizer Prices (yuan/kg):rN=1.30;rK=0.73 Price Ratio(rN/rK)=1.78如果预算更大些，例如400元/公顷，那么，购买225公斤/公顷的N肥和150公斤/ 公顷的K肥比较合适；如果有资金650元，那么，购买375公斤/公顷的N肥和225公斤/公顷的K肥比较合适。If the budget was larger - for example, the capital available was

53、 400 yuan per hectare - then the optimal purchase would be 225 kg/ha of nitrogen and 150 kg/ha of potash. Finally, if the capital available was 650 yuan, then the optimal purchase would be 375 kg/ha of nitrogen and 225 kg/ha of potash.有限制因素的最后一种例子是，假定有两种作物，每种作物需要不同的肥料。具体地说，第一种作物需要N肥，第二种作物需要K肥。如果购买化肥

54、的资金有限，那应当如何分派这笔有限的资金呢？As a final example of a constraint, suppose there are two crops, and that each crop responds to a different fertilizer. More specifically, assume that crop 1 responds to nitrogen, while crop 2 responds to potash. If there is a constraint in the amount of yuan that can be spent

55、on fertilizer, how should this limited amount of yuan be allocated?假定多一元的资金用于第一种作物需要的N肥上，而不是第二种作物需要的K肥上，那就可以多购买1/rN公斤N肥，多生产MPP1/rN公斤的第一种作物，其价值为P1.MPP1/rN，这是多用一元钱在N肥上所产生的边际收益。Suppose an additional yuan is spent on nitrogen on crop 1, rather than on potash on crop 2. With one more yuan, an addition 1/

56、rN kilograms of nitrogen can be purchased. This results in an additional MPP1/rN kilograms of crop 1 output. The value of this additional output is p1MPP1/rN. This is the marginal benefit of spending an additional yuan on nitrogen.由于这一元钱用于N肥，就不能再买1/rK公斤的K肥，这意味着相称于MPP2/rK公斤的第二种作物就被放弃了，其价值为P2.MPP2/rK，

57、这是机会成本。As a result of spending this yuan on nitrogen, 1/rK kilograms of potash can no longer be purchased, which means that output equal to MPP2/rK kilograms of crop 2 is given up. The value of this extra output is p2MPP2/rK. This is the opportunity cost.机会成本等于边际收益，便得出了最大利润的体现式。也就是说，当利润达到最大时P1?MPPN/

58、rN=P2.MPPK/rK。Equating the opportunity cost to the marginal benefit gives the expression for maximizing profits. That is, profits will be maximized whenP1MPPN/rN=P2MPPK/rK图4表达了如何应用这一原则来决定资金在用于两种作物的两种肥料之间的分派。第一幅图表达使用在第一种作物上的N肥的边际产值与使用在N肥上资金的关系；第二幅图表达使用在第二种作物上的K肥的边际产值与使用在K肥上资金的关系。Figure 4 shows how th

59、is rule can be applied to determine the allocation of capital between the two fertilizers on the two crops. The first panel of Figure 4 graphs the MVP of nitrogen fertilizer on crop 1 versus the expenditure on nitrogen required to produce that MVP. The second panel graphs the MVP of potash fertilize

60、r on crop-2 versus the expenditure on potash required to produce that MVP.(图：图4 有限的资金在两种作物之间的分派)图5是图4中第一图情形的细化。如果产生边际效益MVPN所需要的N肥量为XN，那么在N肥价格为rN的状况下，产生边际效益MVPN所需要的资金便为rN.XN。Figure 5 shows the first panel in more detail. For instance, if a quantity XN of fertilizer is required to produce a marginal v

61、alue product of MVPN, then with a price of nitrogen of rN, the expenditure required to produce marginal value product MVPN will be rNXN.(图：图5 边际价值产品（MVP）与肥料费用之间关系的拟定)图4的第三幅图表达，在两种作物中产生相似水平的边际产值需要投在化肥上的总资金，其曲线为第一、二图曲线的水平相加。The third panel of Figure 4 shows the total expenditure on fertilizer required

62、 to generate the same level of marginal value product in both crop 1 and crop 2. The curve in the third panel is drawn by horizontally adding together the curves in the first two panels.假定可运用的资金总量为Yuanonstraint，其中，YuanNconstraint应分派给第一种作物，YuanKconstraint分派给第二种作物，由于YuanNconstraint+YuanKconstraint=Yua

63、nconstraint，因此所有资金都得到了使用。Suppose the total amount of yuan available is Yuanonstraint. Of this, an amount YuanNconstraint should be allocated to crop 1, while an amount YuanKconstraint should be allocated to crop 2. Since YuanNconstraint+YuanKconstraint=Yuanconstraint, all of the available capital is used.可运用肥料的限制因素?-实例 Constraints in the Availability of Fertilizer - Examples上一讲中用了湖南省在中东部低产区农民受到K肥限制的资料。如表3所示。在低产区44名被采访的农民中，25名没使用K肥，这可以阐明这个地区有相称一部分农民得不到钾肥。 In Ecnomics of Fertilizer , I presented data that showed farmers in the low yield region of east-centr