试谈存在限制因素情况下的利润优化

肥料经济学（）-存在限制因素情况下的利润优化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在肥料经济学的第一讲中，我阐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，我说明了边际报酬递减就是边际产量随着肥料用量的增加而递减。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 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取得最高产量。然而，最高产量通常不是适宜的目标。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方法是利润。 In 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 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，当这个比例确定时，肥料的使用量要低于取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 fertilizer price/output price ratio. When this is done, the amount of fertilizer used is less than that used to maximize production.边际产量应该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，现实情况通常并非如此。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，由于缺乏信贷，农民同样受到资金不足的限制，此外，还可能受到化肥生产量或进口量不足的限制。The argument 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 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 constrained 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情况下，如何使用肥料以达到最大利润。首先讨论如何将有限的肥料在各种作物之间进行分配，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配，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. First, 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.我们以介绍基本概念作为讨论这一主题的开始，以使这个问题易于理解。其中最主要的是机会成本概念。首先，我要给机会成本下个定义，然后再用这一概念来分析怎样在不同作物之间对肥料进行分配，以及怎样在不同肥料之间对资金进行分配。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 opportunity 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说明这些概念是如何应用的。After I have outlined the conceptual 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关键。One of the most important concepts in economics is opportunity cost. As I will show, this concept is key to answering the questions I have just posed.给机会成本下定义最好从问题入手。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它作物的机会。Opportunity cost is best defined with the help of a question. What is the cost of using nitrogen fertilizer on a rice crop? 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 is the loss of the opportunity of using it on a different crop.机会成本不过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，因为这时氮肥用于水稻并不涉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 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.限制因素的存在说明了资源的短缺。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配的肥料量有限，肥料就会不足。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，限制因素涉及到机会成本。确定这个机会成本是确定如何分配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.可利用肥料的限制因素理论分析 Constraints in the Availability of Fertilizer - Theory肥料限制最简单的一种情形是生产一种作物，而且这种作物只需要一种肥料。在这种情况下，决策标准很简单：如果可供利用的肥料量小于使利润达到最大值所需要的肥料量，则使用全部肥料。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 available is less than the amount that would maximize profits, then use all the fertilizer.(图：图1 肥料量有限情况下的利润优化)图1说明了这种情况，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限制值小于K*，那么农民应该用K constraint。当然，如果K constraint限制值大于K*，那么只该用K*，余下的部分应贮存到明年，或转售他人。Figure 1 illustrates this case. The level of fertilizer that 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还表明，如果能以某种方式消除限制因素，就会增加收益。例如，如果可供使用的肥料增加一个单位，利润便会增加。利润的增加量为利润曲线在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. The 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都表示在图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 the 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.这里应该说明一下，本讲座随后将用P表示产品价格，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来表示，以此类推。 I should point out here that in the remainder of this lecture, p will be used to refer to the 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大于r，所以，如果能以价格r多购买一个单位的肥料，利润便会增加。这是因为，肥料单位用量的增加使产量增加了MPP。这个增加产量的价值为MVP（MVPP.MPP），获得这个边际收益的成本为r，即肥料的价格。由于MVP大于r，因而收益大于成本，利润增加，利润的增加量为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 output 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利于理解更复杂的问题。例如，假定现在有两种作物，两种作物都对同一种肥料有反应，如果肥料量有限，应该如何在这两种作物之间分配这种肥料呢？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 fertilizer be allocated between the two crops?这个问题的关键是想想有限的肥料量最初在两种作物之间的分配，这表示在图2中。图中的矩形表示固定的肥料量，虚线表示肥料在两种作物之间的分配。现在考虑把一公斤肥料从一种作物转移到另一种作物的结果。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 graphically. 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的产量增加MPP2。从产值上看，作物1的利润降低MVP1=P1.MPP1，作物2的利润增加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 reduces 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大于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，随着肥料转移的增加，MVP2和MVP1之间的差异将变小。因为用于第二种作物肥料的增加使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 fertilizer 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加。To maximize profits, fertilizer should be shifted between the two crops 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它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 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, 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用图示表示了这个问题。第一幅图表示第一种作物各个边际产值水平所需要的肥料量，第二幅图表示在第二种作物上同样的情况，第三幅图表示在第一、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。 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 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配)假定可用的钾肥总量为Kconstraint，其中K1constraint限制量应该用于第一种作物，K2constraint应该用于第二种作物，由于K1constraint+K2constraint=Kconstraint，所有的肥料均得到使用。另外，由于钾肥是在MVP1=MVP2的情况下进行分配的，任何肥料的重新分配都将导致利润下降。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=Kconstraint, 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它有限制的问题。例如，假定某种作物（如水稻）需要两种肥料（氮和钾），而且这两种肥料相互作用以决定产量，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配呢？The notion of opportunity 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 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，从而使水稻产量增加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 more 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），其结果是水稻产量下降MPPK/rK公斤，这就是机会成本。As a result of spending this yuan on nitrogen, 1/rK kilograms of potash 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表达式，即当利润最大时MPPN/rN=MPPK/rK这个表达式也可改写为：MPPN/MPPK=rN/rK Equating the opportunity cost to the marginal benefit gives the expression for maximizing profits. That is, profits will be maximized whenMPPN/rN=MPPK/rKwhich is equivalent toMPPN/MPPK=rN/rK表1用一个假想的例子说明了这一原则。表的上半部分表示在不同N、K肥水平下N和K的边际产量。N和K肥均表现出边际报酬递减，N肥的边际产量随着N肥的增加而下降，K肥的边际产量也随着K肥的增加而减少。表的下半部分表示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 applied 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 the 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 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肥的价格比大约为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 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配 )N (kg/ha) 施氮量 氮钾边际产量比 MPPN/MPPK 花费 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 3.6 347 402 457 300 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，比如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 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个例子是，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。如果购买化肥的资金有限，那应该如何分配这笔有限的资金呢？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 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/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，这意味着相当于MPP2/rK公斤的第二种作物就被放弃了，其价值为P2.MPP2/rK，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表达式。也就是说，当利润达到最大时P1?MPPN/rN=P2.MPPK/rK。Equating the opportunity cost to the marginal benefit gives the expression for maximizing profits. That is, profits will be maximized