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ACCA PM定价：The algebraic approach

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发布时间：2021-08-13 16:57

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The algebraic approach

The problem with the tabular approach used above is that it only looked at the data in increments of 1,000 units.That allowed us to conclude that profit was maximised when 3,000 units were sold.But could profits be better if,say,sales were 2,900,3,010 or 3,100 units?We have no information in the table that allows the advice to be refined.

To obtain greater accuracy instead of relying on tables,the relationship between quantity and price has to be described by an equation.The equation will be of the form:

P=a-bQ

Where P=price per unit and Q=quantity sold.a and b are constants that have to be found using the demand curve data.This formula is supplied in the PM exam.

We know from the table that at a price of(say)$140,2,000 units are demanded.So:

140=a–b x 2,000

Moving to the previous quantity level,when Q=1,000,P=160.So:

160=a–b x 1,000

a and b are then found by simultaneous equations.Subtracting the two equations gives:

-20=-b x 1,000

So b=20/1,000=0.02

Put that information into either of the two original equations to find a.So the first equation becomes:

140=a–0.02 x 2,000=a–40

a=180.

The demand curve is therefore

P=180–0.02Q

Note that the formula sheet gives a simple way of finding a and b if the required information is supplied:

a=price when Q=0.From the table this is 180

b=change in price/change in quantity.For each increment of the table price changes by 20 and the quantity changes by 1,000,so b=20/1,000=0.02

The formula sheet also shows how to convert the demand curve equation to an equation for the marginal revenue.

Demand curve:P=a–bQ P=180–0.02Q

MR:MR=a–2bQ MR=180–0.04Q

We know that the marginal cost per unit is$65,the variable cost per unit.Profit is maximised when

MC=MR

65=180–0.04Q

Solving for Q:

-115=-0.04Q

115=0.04Q

Q=115/0.04=2,875

You will see that this is slightly less than our previous answer of 3,000.What’s happened is that as output increases from 2,000 to 3,000,the early additional units(2,001,2,002…etc)are worthwhile but later ones are not(2,997,2998,2,999,3,000).However,all of this detail is lost and in total moving from 2,000 to 3,000 is worthwhile.Moving from 2,000 to 2,875 is even better.

Now used this quantity figure in the original demand curve equation to work out the selling price.

P=180–(0.02 x 2,875)

P=122.5

So,using the algebraic approach,profits will be maximised when 2,750 units are sold and these sales will be generated if the selling price is set at$122.50.

The second article on pricing will deal with more practical aspects of pricing decisions.

Written by Ken Garrett,a freelance lecturer and writer

翻译参考

代数方法

上面使用的表格方法的问题在于它只查看以1,000个单位为增量的数据。这使我们得出结论，当售出3,000台时，利润最大化。但是，如果销售量为2,900、3,010或3,100台，利润会不会更好？我们在表格中没有可以改进建议的信息。

为了获得更高的准确性而不是依靠表格，数量和价格之间的关系必须用一个方程来描述。该等式将采用以下形式：

P=a-bQ

其中P=每单位价格，Q=销售数量。a和b是必须使用需求曲线数据找到的常数。这个公式在PM考试中提供。

我们从表中知道，在（比如）140美元的价格下，需要2,000件。所以：

140=a–bx 2,000

移动到之前的数量级别，当Q=1,000，P=160时。所以：

160=a–bx 1,000

然后通过联立方程找到a和b。两个方程相减得到：

-20=-bx 1,000

所以b=20/1,000=0.02

将该信息放入两个原始方程中的任何一个中以找到a。所以第一个方程变为：

140=a–0.02 x 2,000=a–40

a=180。【点击免费下载>>>更多ACCA学习相关资料】

因此需求曲线为

P=180–0.02Q

请注意，如果提供所需信息，公式表提供了一种简单的方法来查找a和b：

a=Q=0时的价格。从表中可以看出这是180

b=价格变化/数量变化。表价格每增加20次，数量增加1,000次，所以b=20/1,000=0.02

公式表还显示了如何将需求曲线方程转换为边际收益方程。

需求曲线：P=a–bQ P=180–0.02Q

MR：MR=a–2bQ MR=180–0.04Q

我们知道每单位的边际成本是65美元，即每单位的可变成本。利润最大化时

MC=MR

65=180–0.04Q

求解Q：

-115=-0.04Q

115=0.04Q

Q=115/0.04=2,875

您会看到这比我们之前的答案3,000略少。实际情况是，随着产量从2,000增加到3,000，早期的额外单位（2,001、2,002等）是值得的，但后来的则不值得（2,997、2998、2,999、3,000）。然而，所有这些细节都丢失了，从2,000移动到3,000是值得的。从2,000移动到2,875甚至更好。

现在在原始需求曲线方程中使用这个数量数字来计算销售价格。

P=180–(0.02 x 2,875)

P=122.5

因此，使用代数方法，当销售2,750件时，利润将最大化，如果售价设定为122.50美元，就会产生这些销售额。

关于定价的第二篇文章将涉及定价决策的更多实际方面。

【ACCA PM定价：Demand curves】

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