Here is an example from cost accounting:
In Chapter 15 of Cost Accounting: A Managerial Emphasis, 14th ed., by Horngren, Datar, and Rajan, one of the sections covers a method of allocating costs of support departments such as Personnel and Legal to operating departments such as the refrigerator and dish washer manufacturing divisions of a kitchen appliances manufacturer. Special consideration needs to be given for the common scenario where support departments support each other. For example, the Personnel department may support the Legal department by hiring paralegals and attorneys while the Legal department might support the Personnel department by evaluating compensation plans and interpretting regulatory requirements.
The most justifiable method presented in the chapter is called the "reciprocal method" or "matrix method". It uses a system of linear equations to calculate the total cost of each support department, factoring in reciprocal support, so that the total cost can be allocated to the operating departments.
For example, assume the following facts:
- The Personnel Department's budgeted fixed costs are $1,200,000, not including reciprocal support costs.
- The Legal Department's budgeted fixed costs are $2,000,000, not including reciprocal support costs.
- Personnel Department costs are allocated on the basis of recruiter hours.
- Legal Department costs are allocated on the basis of attorney hours.
- The Personnel Department is budgeted to utilize 5% of the Legal Department's budget for attorney hours.
- The Legal Department is budgeted to utilize 10% of the Personnel Department's budget for recruiter hours.
Let $P_F$ be the Personnel Department's total fixed costs and $L_F$ be the Legal Department's total fixed costs. Then:
P_F &= \$1,200,000 + 5\% \times L_F\\
L_F &= \$2,000,000 + 10\% \times P_F
L_F &= \$2,000,000 + 10\% \times (\$1,200,000 + 5\% \times L_F) \\
99.5\% \times L_F &= \$2,120,000 \\
L_F &= \$2,130,653.27
P_F &= \$1,200,000 + 5\% \times \$2,130,653.27\\
The related matrix is:
1 & -0.05 \\
-0.1 & 1
The system of linear equations is larger with more support departments.