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== Explicit formal analysis of the multi-product case ==
== Examples and formal analysis ==
 
Usually, the different approaches result in different break-even-points (or sets of break-even-points). This can be shown by [[Accounting exercises 4: Break-even-Analysis|examples]] or by [[Formal_multi-product_break-even-analysis|formal multi-product break-even-analyses]].
Simplification is a useful way to deal with complex problems in many situations but sometimes simple approaches neglect critical information and exclude useful solutions. A more sophisticated solution can make the difference in a competitive environment. At least, the more sophisticated solution serves as a benchmark and allows both quantitative and qualitative assessments of other approaches. We introduce the explicit analysis of the multi-product case in the following steps:
* Analysis of the case of two products with joint fixed costs
* Analysis of the case of two products with joint and individual fixed costs
* Analysis of the two-product case with restrictions
* Analysis of the multi-product case
* Analysis of the multi-product case with restrictions.
 
 
=== The case of two products with joint fixed costs ===
 
If we know the constant unit price (p<sub>1</sub>, p<sub>2</sub>) and unit variable cost (k<sup>v</sup><sub>1</sub>, k<sup>v</sup><sub>2</sub>) for two products, and the joint fixed costs (K<sup>f</sup>) necessary for their production, we can put the break-even-condition with respect to the production volumes (x<sub>1</sub>, x<sub>2</sub>) in the following way:<br>
E = K = p<sub>1</sub> &middot; x<sub>1</sub> + p<sub>2</sub> &middot; x<sub>2</sub> = k<sup>v</sup><sub>1</sub> &middot; x<sub>1</sub> + k<sup>v</sup><sub>2</sub> &middot; x<sub>2</sub> + K<sup>f</sup><br>
Using unit contributions d<sub>1</sub> and d<sub>2</sub> respectively, and rearranging results in:<br>
K<sup>f</sup> = (p<sub>1</sub>-k<sup>v</sup><sub>1</sub>) &middot; x<sub>1</sub> + (p<sub>2</sub>-k<sup>v</sup><sub>2</sub>) &middot; x<sub>2</sub> = d<sub>1</sub> &middot; x<sub>1</sub> + d<sub>2</sub> &middot; x<sub>2</sub><br>
It is obvious that the necessary contribution of one product depends on the contribution provided by the other product. In extreme cases, we concentrate on one of the products and set the volume of the other equal to zero. This leads to the individual break-even-points (or break-even-corner-points):<br>
x<sub>1</sub>* = K<sup>f</sup>/d<sub>1</sub> (if x<sub>2</sub> = 0)<br>
x<sub>2</sub>* = 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 x<sub>1</sub>* and x<sub>2</sub>* will cover the fixed costs too. This can be easily seen for x<sub>1</sub> = .5* x<sub>1</sub>* and x<sub>2</sub> = .5 &middot; x<sub>2</sub>*: <br>
d<sub>1</sub> &middot; x<sub>1</sub> + d<sub>2</sub> &middot; x<sub>2</sub> = d<sub>1</sub> &middot; .5 &middot; x<sub>1</sub>* + d<sub>2</sub> &middot; .5 &middot; x<sub>2</sub>* = d<sub>1</sub> &middot; .5 &middot; K<sub>2</sub>/d<sub>1</sub> + d<sub>2</sub> &middot; .5 * K<sup>f</sup>/d<sub>2</sub> = K<sub>2</sub>.<br>
Other linear combinations allow fixed-cost-coverage in the same way. Together, they 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>
 
 
 
 
t.b.c.
 
 
[[Break-even-Analyse]]




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