# How do you apply the reciprocal allocation method?

The reciprocal allocation method is a method of allocating support department (or activity) costs which accounts for services provided both to operating departments and to other support departments. This method explicitly includes all mutual services rendered among support departments.

This method requires first expressing support department costs and reciprocal relationships in a system of linear equations. Knowing that 25,000 euros were initially accumulated in HR and 19,000 euros in maintenance and building on the diagram above, you get the following system:

$\left\{ \begin{array}{ll} HR = 25,000 + 0.25 \times M \\ M = 19,000 + 0.5 \times HR \\ \end{array} \right.$

which you can read that way: the allocation from HR (“HR”) is equal to the amount initially accumulated in HR (25,000) plus the amount allocated from maintenance, so 25% of the allocation from maintenance (“M”); the allocation from maintenance (“M”) is equal to the amount initially accumulated in maintenance (19,000) plus the amount allocated from HR, so 50% of the allocation from HR (“HR”). The “allocation” is not the amount initially accumulated, but the total cost of providing each service, including the cost incurred in other support department. This allocation is also called complete reciprocated cost.

The complete reciprocated cost is the cost incurred by a focal service department, taking into account the costs allocated from other service departments because of the services they provided to this focal service department.

I will now illustrate three ways to solve this system of equations:

• by substitution;
• through manual matrix algebra;
• with Excel.

Substitution

You can get all the data and solutions by downloading the same Excel as before (I only put this link again in case you missed it the first time). Substitution consists in replacing any unknown in any one equation by the other equation:

\begin{aligned} HR & = 25,000 + 0.25 \times M \\ HR & = 25,000 + 0.25 \times (19,000 + 0.5 \times HR) \\ HR & = 25,000 + 4,750 + 0.125 \times HR \\ HR - 0.125 \times HR & = 25,000 + 4,750 \\ 0.825 \times HR & = 29,750 \\ HR & = \frac{29,750}{0.875} = 34,000 \\ \quad \\ M & = 19,000 + 0.5 \times HR \\ M & = 19,000 + 0.5 \times 34,000 \\ M & = 36,000 \end{aligned}

Then you divide the allocation or complete reciprocated amount but the total quantity of allocation base and you obtain the allocation rate you can use for allocation.

Matrix algebra

In matrix algebra, you transform the system in a product of matrices. To do this, first reorganize the system with all the unknowns in the same order on the left and the known on the right:

\left\{ \begin{array}{ll} \begin{aligned} 1 & \times && HR &&& - 0.25 \times &&&& M &&&&& = 25,000 \\ -0.5 & \times && HR &&& + 1 \times &&&& M &&&&& = 19,000 \\ \end{aligned} \end{array} \right.

Leading to the following matrix:

$\begin{pmatrix} 1 & -0.25 \\ -0.5 & 1 \end{pmatrix} \begin{pmatrix} HR \\ M \end{pmatrix} = \begin{pmatrix} 25,000 \\ 19,000 \end{pmatrix}$

Which can be transformed into the following by multiplying the second line by 2:

$\begin{pmatrix} 1 & -0.25 \\ -1 & 2 \end{pmatrix} \begin{pmatrix} HR \\ M \end{pmatrix} = \begin{pmatrix} 25,000 \\ 38,000 \end{pmatrix}$

Then adding the two rows:

$\begin{pmatrix} 0 & 1.75 \end{pmatrix} \begin{pmatrix} HR \\ M \end{pmatrix} = \begin{pmatrix} 63,000 \end{pmatrix}$

Which translates into:

\begin{aligned} 0 \times HR + 1.75 \times M & = 63,000 \\ M & = \frac{63,000}{1.75} = 36,000 \end{aligned}

Excel

The reciprocal method is the most complex and therefore the most expensive allocation method, but it is also the most accurate. The allocation rate obtained with this method is the best available approximation of the cost of providing a specific service. It takes into account not only the cost of the resources visibly consumed while providing the service, but also the cost of all the supporting activities which are necessary for this service to be delivered.

This is important because allocation rates also serve the very important purpose of making managers aware of the cost of the internal services they use. The underlying reasoning is that if they do not pay for it, they may ask for more than necessary. However, if they are aware of what it costs to consume a service, there will consume it only if they think that they can get a greater benefit out of this consumption.

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