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Formal multi-product break-even-analysis (Quelltext anzeigen)
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''by Clemens Werkmeister'' | ''by Clemens Werkmeister'' | ||
<html><img src= | <html><img src="http://vg06.met.vgwort.de/na/c7a3898531b948bb8f88a959a26e3156" width="1" height="1" alt=""></html> | ||
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A detailed analysis of theses cases is available in Schweitzer/Troßmann (1998) and Schweitzer/Troßmann/Lawson (1992). The [[Accounting exercises 4: Break-even-Analysis|exercise pages]] offer useful examples.<br> | A detailed analysis of theses cases is available in Schweitzer/Troßmann (1998) and Schweitzer/Troßmann/Lawson (1992). The [[Accounting exercises 4: Break-even-Analysis|exercise pages]] offer useful examples.<br> | ||
== The case of two products with joint fixed costs == | == The case of two products with joint fixed costs == | ||
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: [[Datei:x_hat_2.png]] = K<sup>f</sup>/d<sub>2</sub> (if x<sub>1</sub> = 0)<br> | : [[Datei:x_hat_2.png]] = K<sup>f</sup>/d<sub>2</sub> (if x<sub>1</sub> = 0)<br> | ||
Since we assumed '''linear cost-volume-profit-relations''' (due to the constant unit prices and variable costs), a 50/50-combination of | Since we assumed '''linear cost-volume-profit-relations''' (due to the constant unit prices and variable costs), a 50/50-combination of [[Datei:x_hat_1.png]] and [[Datei:x_hat_2.png]] will cover the fixed costs too. This can be easily seen for x<sub>1</sub> = 0.5 · [[Datei:x_hat_1.png]] and x<sub>2</sub> = 0.5 · [[Datei:x_hat_2.png]]: <br> | ||
: d<sub>1</sub> · x<sub>1</sub> + d<sub>2</sub> · x<sub>2</sub> = d<sub>1</sub> · 0.5 · | : d<sub>1</sub> · x<sub>1</sub> + d<sub>2</sub> · x<sub>2</sub> = d<sub>1</sub> · 0.5 ·[[Datei:x_hat_1.png]] + d<sub>2</sub> · 0.5 · [[Datei:x_hat_2.png]] = d<sub>1</sub> · 0.5 · K<sup>f</sup>/d<sub>1</sub> + d<sub>2</sub> · 0.5 · K<sup>f</sup>/d<sub>2</sub> = K<sup>f</sup>.<br> | ||
Other linear combinations enable fixed-cost-coverage in a similar way. Together, the corresponding break-even-points build a straight line between the break-even-corner-points, the '''break-even-line,''' as can be seen in the following three-dimensional graph (with x<sub>1</sub> and x<sub>2</sub> as horizontal axes and the profit, contribution and costs at the vertical axis.<br> | Other linear combinations enable fixed-cost-coverage in a similar way. Together, the corresponding break-even-points build a straight line between the break-even-corner-points, the '''break-even-line,''' as can be seen in the following three-dimensional graph (with x<sub>1</sub> and x<sub>2</sub> as horizontal axes and the profit, contribution and costs at the vertical axis.<br> | ||
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In order to facilitate algebraic formulations, we introduce the price vector (p<sub>1</sub>, p<sub>2</sub>), the variable cost vector (k<sup>v</sup><sub>1</sub>, k<sup>v</sup><sub>2</sub>), the unit contribution vector (d<sub>1</sub>, d<sub>2</sub>) and the output vector (x<sub>1</sub>, x<sub>2</sub>). The break-even-line is defined by the following output vector: | In order to facilitate algebraic formulations, we introduce the price vector (p<sub>1</sub>, p<sub>2</sub>), the variable cost vector (k<sup>v</sup><sub>1</sub>, k<sup>v</sup><sub>2</sub>), the unit contribution vector (d<sub>1</sub>, d<sub>2</sub>) and the output vector (x<sub>1</sub>, x<sub>2</sub>). The break-even-line is defined by the following output vector: | ||
: [[Datei:break-even-line-set.png]] | : [[Datei:break-even-line-set.png]]<br> | ||
==The case of two products with individual and joint fixed costs== | ==The case of two products with individual and joint fixed costs== | ||
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: [[Datei:break-even-line-joint-set.png]] | : [[Datei:break-even-line-joint-set.png]] | ||
Note, that different to the previous case without individual fixed costs (and different to the formula proposed in Schweitzer/Troßmann (1998, p. 177), in this case the set of break-even-points is not just a linear combination of the two break-even-corner-points[[Datei:x_tilde_1.png]] and [[Datei:x_tilde_2.png]]. Instead, it still is a linear combination of the break-even-points [[Datei:x_hat_1.png]] and [[Datei:x_hat_1.png]], corresponding to the joint fixed costs, plus the units [[Datei:x_circle_1.png]] and [[Datei:x_circle_2.png]] necessary to cover the individual fixed costs. | Note, that different to the previous case without individual fixed costs (and different to the formula proposed in Schweitzer/Troßmann (1998, p. 177), in this case the set of break-even-points is not just a linear combination of the two break-even-corner-points[[Datei:x_tilde_1.png]] and [[Datei:x_tilde_2.png]]. Instead, it still is a linear combination of the break-even-points [[Datei:x_hat_1.png]] and [[Datei:x_hat_1.png]], corresponding to the joint fixed costs, plus the units [[Datei:x_circle_1.png]] and [[Datei:x_circle_2.png]] necessary to cover the individual fixed costs.<br> | ||
== Analysis of the two-product case with restrictions== | == Analysis of the two-product case with restrictions== | ||
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Using vector notation for the capacity restraint, he set of break-even-points is: <br> | Using vector notation for the capacity restraint, he set of break-even-points is: <br> | ||
: [[Datei:break-even-set-joint-restrictions.png]] | : [[Datei:break-even-set-joint-restrictions.png]]<br> | ||
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== Evaluation of explicit multi-product break-even-analyses == | == Evaluation of explicit multi-product break-even-analyses == | ||
A large number of managerial decisions deal with multi-product or similar problems (for example product or regional market differentiation, or aiming at economies of scope). They combine both individual and joint production processes, and hence, cost. In many cases, the exact outputs of those products or volumes of the markets are unknown. Thus, a break-even- model that provides information about critical outputs or market volumes, is useful. Break-even-corner-points and intuitive graphical representations of the cost-volume-profit-problem support the decision making processes and offer valuable additional information compared to the simplifying use of single-product or single-index break-even approaches. However, the more products we include in the analysis, the more contingent are our break-even-hyper-planes. Whether a certain quantity of a product is sufficient or not, depends on the outputs of (too) many other products. In such situations, other approaches may provide better guidelines for decision-making. Apart from the simplifying break-even-analyses already mentioned, this could be the identification of typical output combinations and the calculation of their profit. | A large number of managerial decisions deal with multi-product or similar problems (for example product or regional market differentiation, or aiming at economies of scope). They combine both individual and joint production processes, and hence, cost. In many cases, the exact outputs of those products or volumes of the markets are unknown. Thus, a break-even-model that provides information about critical outputs or market volumes, is useful. Break-even-corner-points and intuitive graphical representations of the cost-volume-profit-problem support the decision making processes and [[Controlling|controlling]] and offer valuable additional information compared to the simplifying use of single-product or single-index break-even approaches. However, the more products we include in the analysis, the more contingent are our break-even-hyper-planes. Whether a certain quantity of a product is sufficient or not, depends on the outputs of (too) many other products. In such situations, other approaches may provide better guidelines for decision-making. Apart from the simplifying break-even-analyses already mentioned, this could be the identification of typical output combinations and the calculation of their profit.<br> | ||