<p>Writing out the formulas for department <span class="math-container">$1$</span> and <span class="math-container">$2$</span> gives:
<span class="math-container">$$
\begin{align*}
CPMH_1 &amp;= \left(\frac{LCB_1}{WHB_1} - \frac{LCA_1}{WHA_1}\right)*WHA_1\\
&amp;= \frac{LCB_1}{WHB_1}WHA_1 - LCA_1\\
CPMH_2 &amp;= \left(\frac{LCB_2}{WHB_2} - \frac{LCA_2}{WHA_2}\right)*WHA_2\\
    &amp;= \frac{LCB_2}{WHB_2}WHA_2 - LCA_2
\end{align*}
$$</span>
For both departments together and substituting out we get:
<span class="math-container">$$
\begin{align*}
CPMH &amp;=  \left(\frac{LCB_1+ LCB_2}{WHB_1 + WHB_2} - \frac{LCA_1 + LCB_2}{WHA_1 + WHA_2}\right)(WHA_1 + WHA_2)\\
&amp;= \frac{LCB_1 + LCB_2}{WHB_1 + WHB_2}(WHA_1 + WHA_2) - LCA_1 - LCB_2,\\
&amp;= \frac{LCB_1 + LCB_2}{WHB_1 + WHB_2}(WHA_1 + WHA_2) \\
&amp;+ CPMH_1 - \frac{LCB_1}{WHB_1} WHA_1 + CPMH_2 - \frac{LCB_2}{WHB_2}WHA_2 
\end{align*}
$$</span>
So:
<span class="math-container">$$
\begin{align*}
&amp;CPMH - CPMH_1 - CPMH_2\\
&amp;= \frac{LCB_1 + LCB_2}{WHB_1 + WHB_2}(WHA_1 + WHA_2) - \frac{LCB_2}{WHB_2} WHA_2 - \frac{LCB_1}{WHB_1} WHA_1,\\
&amp;= LCB_2\left(\frac{WHA_1 + WHA_2}{WHB_1+ WHB_2} - \frac{WHA_2}{WHB_2}\right)
+ LCB_1\left(\frac{WHA_1 + WHA_2}{WHB_1 + WHB_2} - \frac{WHA_1}{WHB_1}\right)
\end{align*}
$$</span>
This will usually not be equal to zero unless:
<span class="math-container">$$
\frac{WHA_1 + WHA_2}{WHB_1 + WHB_2} = \frac{WHA_2}{WHB_2} = \frac{WHA_1}{WHB_1}
$$</span></p>

CC BY-SA 4.0
2022-03-23T15:31:30.653Z
2022-03-23T15:31:30.653Z
tdm
<p>There's nothing wrong with your formulas. It's just that the CPMH impact is not a linear function and therefore the CPMH impact of a total is not the sum of the CPMH impacts of its parts.</p>
<p><strong>ADDED after your EDIT:</strong> From a conceptual perspective, some measures of &quot;output&quot;, &quot;productivity&quot;, &quot;costs&quot;, &quot;impact&quot;, or whatever, simply &quot;add up&quot; from the department level to the company level, while others don't. E.g. <em>number of workers</em> adds up, <em>working hours</em> adds up, <em>wage expenditures</em> adds up, but <em>wage expenditures per worker</em> or <em>output per working hour</em> doesn't add up. This is of course because the latter are (weighted) averages, and averages don't simply add up. E.g. if both departments pay wages of \$10 per hour, this doesn't mean that the company pays \$20 per hour. Now, your <em>CPMH impact</em> is a more complicated measure, but it contains something similar to an average, and as a consequence it simply doesn't add up. Your intuition that &quot;financial impact&quot; should add up might be correct for a simple measure of &quot;financial impact&quot; like <em>revenue</em>, but for your rather involved CPMH measure it is not.</p>

CC BY-SA 4.0
2022-03-23T15:24:57.303Z
2022-03-24T11:33:44.817Z
2022-03-24T11:33:44.817Z
VARulle
<p>In a company with two departments, we know the actual &amp; budget of working hours (WH) and the labor cost (LC) of both of them. I want to compute the cost per men-hour (CPMH) and the economic impact of the actual-budget difference. Thus, I have the following formulas:</p>
<p><span class="math-container">$\text{CPMH actual} = \frac{\text{LC actual}}{\text{WH actual}}$</span></p>
<p><span class="math-container">$\text{CPMH budget} = \frac{\text{LC budget}}{\text{WH budget}}$</span></p>
<p><span class="math-container">$\text{CPMH Impact} = (\text{CPMH budget} - \text{CPMH actual}) * \text{WH actual}$</span></p>
<p>Applying real data to it, this is how this computation looks like:</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th style="text-align: center;"></th>
<th style="text-align: center;">WH ACT</th>
<th style="text-align: center;">WH BUD</th>
<th style="text-align: center;">LC ACT</th>
<th style="text-align: center;">LC BUD</th>
<th style="text-align: center;">CPMH ACT</th>
<th style="text-align: center;">CPMH BUD</th>
<th style="text-align: center;">CPMH impact</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: center;">Dpmt 1</td>
<td style="text-align: center;">1800 h</td>
<td style="text-align: center;">2016 h</td>
<td style="text-align: center;">€  32,121.00</td>
<td style="text-align: center;">€  40,320.00</td>
<td style="text-align: center;">17.85 €/h</td>
<td style="text-align: center;">20.00 €/h</td>
<td style="text-align: center;"><strong>€  3,879.00</strong></td>
</tr>
<tr>
<td style="text-align: center;">Dpmt 2</td>
<td style="text-align: center;">1910 h</td>
<td style="text-align: center;">1890 h</td>
<td style="text-align: center;">€  48,168.00</td>
<td style="text-align: center;">€  42,525.00</td>
<td style="text-align: center;">25.22 €/h</td>
<td style="text-align: center;">22.50 €/h</td>
<td style="text-align: center;"><strong>-€ 5,193.00</strong></td>
</tr>
<tr>
<td style="text-align: center;"><strong>TOTAL</strong></td>
<td style="text-align: center;">3710 h</td>
<td style="text-align: center;">3906 h</td>
<td style="text-align: center;">€  80,289.00</td>
<td style="text-align: center;">€  82,845.00</td>
<td style="text-align: center;">21.64 €/h</td>
<td style="text-align: center;">21.21 €/h</td>
<td style="text-align: center;"><strong>-€ 1,601.10</strong></td>
</tr>
</tbody>
</table>
</div>
<p>All the impacts are just computed using the formula. However, if I add <span class="math-container">$3 879€$</span> and <span class="math-container">$-5 193€$</span> the result is <span class="math-container">$-1,314€$</span>, not <span class="math-container">$-1,601.1€$</span>.</p>
<p>Is there something wrong with my formulas?</p>
<p><strong>EDIT:</strong> Thanks <a href="https://economics.stackexchange.com/a/50852/40770">tdm</a> and <a href="https://economics.stackexchange.com/a/50851/40770">VARulle</a> for your answers. We are not wondering <em>how</em> does the formula doesn't add up to the total, but <em>why</em>; from a conceptual rather than mathematical perspective. To me, it seems counter-intuitive to say that breaking the financial impact of our CPMH down into departments does not add up to the overall.</p>

CC BY-SA 4.0
2022-03-24T11:34:59.560Z
O.k., read your edit and added some explanation in my answer below.
VARulle
<mathematical-economics>
Cost per men-hour not adding up
Carlos Roldán
# Cost per men-hour not adding up

(Asked by: username_0 on 2022-03-23)

In a company with two departments, we know the actual & budget of working hours (WH) and the labor cost (LC) of both of them. I want to compute the cost per men-hour (CPMH) and the economic impact of the actual-budget difference. Thus, I have the following formulas:

$\text{CPMH actual} = \frac{\text{LC actual}}{\text{WH actual}}$

$\text{CPMH budget} = \frac{\text{LC budget}}{\text{WH budget}}$

$\text{CPMH Impact} = (\text{CPMH budget} - \text{CPMH actual}) \* \text{WH actual}$

Applying real data to it, this is how this computation looks like:

|  | WH ACT | WH BUD | LC ACT | LC BUD | CPMH ACT | CPMH BUD | CPMH impact |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Dpmt 1 | 1800 h | 2016 h | € 32,121.00 | € 40,320.00 | 17.85 €/h | 20.00 €/h | **€ 3,879.00** |
| Dpmt 2 | 1910 h | 1890 h | € 48,168.00 | € 42,525.00 | 25.22 €/h | 22.50 €/h | **-€ 5,193.00** |
| **TOTAL** | 3710 h | 3906 h | € 80,289.00 | € 82,845.00 | 21.64 €/h | 21.21 €/h | **-€ 1,601.10** |

All the impacts are just computed using the formula. However, if I add $3 879€$ and $-5 193€$ the result is $-1,314€$, not $-1,601.1€$.

Is there something wrong with my formulas?

**EDIT:** Thanks [tdm](https://economics.stackexchange.com/a/50852/40770) and [VARulle](https://economics.stackexchange.com/a/50851/40770) for your answers. We are not wondering *how* does the formula doesn't add up to the total, but *why*; from a conceptual rather than mathematical perspective. To me, it seems counter-intuitive to say that breaking the financial impact of our CPMH down into departments does not add up to the overall.

**Tags:** mathematical-economics
> **Comments:**
> **username_1**: O.k., read your edit and added some explanation in my answer below.  

---

## 2 answers

### Answer by username_2 on 2022-03-23. Score: 3

Writing out the formulas for department $1$ and $2$ gives:
$$
\begin{align\*}
CPMH\_1 &= \left(\frac{LCB\_1}{WHB\_1} - \frac{LCA\_1}{WHA\_1}\right)\*WHA\_1\\
&= \frac{LCB\_1}{WHB\_1}WHA\_1 - LCA\_1\\
CPMH\_2 &= \left(\frac{LCB\_2}{WHB\_2} - \frac{LCA\_2}{WHA\_2}\right)\*WHA\_2\\
&= \frac{LCB\_2}{WHB\_2}WHA\_2 - LCA\_2
\end{align\*}
$$
For both departments together and substituting out we get:
$$
\begin{align\*}
CPMH &= \left(\frac{LCB\_1+ LCB\_2}{WHB\_1 + WHB\_2} - \frac{LCA\_1 + LCB\_2}{WHA\_1 + WHA\_2}\right)(WHA\_1 + WHA\_2)\\
&= \frac{LCB\_1 + LCB\_2}{WHB\_1 + WHB\_2}(WHA\_1 + WHA\_2) - LCA\_1 - LCB\_2,\\
&= \frac{LCB\_1 + LCB\_2}{WHB\_1 + WHB\_2}(WHA\_1 + WHA\_2) \\
&+ CPMH\_1 - \frac{LCB\_1}{WHB\_1} WHA\_1 + CPMH\_2 - \frac{LCB\_2}{WHB\_2}WHA\_2
\end{align\*}
$$
So:
$$
\begin{align\*}
&CPMH - CPMH\_1 - CPMH\_2\\
&= \frac{LCB\_1 + LCB\_2}{WHB\_1 + WHB\_2}(WHA\_1 + WHA\_2) - \frac{LCB\_2}{WHB\_2} WHA\_2 - \frac{LCB\_1}{WHB\_1} WHA\_1,\\
&= LCB\_2\left(\frac{WHA\_1 + WHA\_2}{WHB\_1+ WHB\_2} - \frac{WHA\_2}{WHB\_2}\right)
+ LCB\_1\left(\frac{WHA\_1 + WHA\_2}{WHB\_1 + WHB\_2} - \frac{WHA\_1}{WHB\_1}\right)
\end{align\*}
$$
This will usually not be equal to zero unless:
$$
\frac{WHA\_1 + WHA\_2}{WHB\_1 + WHB\_2} = \frac{WHA\_2}{WHB\_2} = \frac{WHA\_1}{WHB\_1}
$$


---

### Answer by username_1 on 2022-03-23. Score: 2

There's nothing wrong with your formulas. It's just that the CPMH impact is not a linear function and therefore the CPMH impact of a total is not the sum of the CPMH impacts of its parts.

**ADDED after your EDIT:** From a conceptual perspective, some measures of "output", "productivity", "costs", "impact", or whatever, simply "add up" from the department level to the company level, while others don't. E.g. *number of workers* adds up, *working hours* adds up, *wage expenditures* adds up, but *wage expenditures per worker* or *output per working hour* doesn't add up. This is of course because the latter are (weighted) averages, and averages don't simply add up. E.g. if both departments pay wages of \$10 per hour, this doesn't mean that the company pays \$20 per hour. Now, your *CPMH impact* is a more complicated measure, but it contains something similar to an average, and as a consequence it simply doesn't add up. Your intuition that "financial impact" should add up might be correct for a simple measure of "financial impact" like *revenue*, but for your rather involved CPMH measure it is not.


---

