<p>Suppose we want to find the contributions of the variables to the given function:</p>

<p>$$f(x,y)= {x\over{y}}$$</p>

<p>To find the growth rates of the given variables first take the logarithm of the function $f(x,y)$:
$$ln(f(x,y))=ln\left({x\over{y}}\right)=ln(x)-ln(y)$$ </p>

<p>And then take the partial derivative of the variable you would like to analyze:  </p>

<p>$${\partial ln\left({x\over{y}}\right)\over{\partial x}}={\dot{x}\over{x}}$$</p>

<p>$${\partial ln \left({x \over{y}} \right)\over{\partial y}}=-{\dot{y}\over{y}}$$</p>

<p>Where $\dot{x}$ and $\dot{y}$ are the derivatives of $x$ and $y$ respectively</p>

<p>Thus in your case if you would like to find the contributions to growth to leverage you can simply do the same thing as above. </p>

<p>i.e.</p>

<p>$$\mathfrak{L}={{l+b}\over{l+b+e}}$$</p>

<p>Where:</p>

<p>$\mathfrak{L}=$ leverage</p>

<p>$l=$ loans</p>

<p>$b=$ bonds</p>

<p>$e=$ equity</p>

<p>Thus </p>

<p>$$ln \left( \mathfrak{L} \right)= ln(l+b) -ln(l+b+e)$$</p>

<p>you can thus find the contributions to growth by the variables involved by taking the partial derivative of the variable you wish to examine. </p>

<p>Hope this is helpful!!  </p>

CC BY-SA 3.0
2018-03-04T19:09:48.897Z
2018-03-04T19:09:48.897Z
Bensstats
<p>I have a micro dataset of firms' financial liabilities for 15 years. I would like to plot the contributions to change in leverage, decomposing the single contribution of bank loans, bonds and equity variations.
Leverage is built as:</p>

<pre><code>Leverage=(Loans+Bonds)/(Loans+Bonds+Equity)
</code></pre>

<p>Therefore contributions to growth are positive if financial debt (Bank Loans or Bonds) increases or if its sum with capital decreases and vice versa.</p>

<p>I would be able to decompose with logarithm a simple ratio a/b but I have no clue on how to do it with 3 components. Any idea? Thanks</p>

<microeconomics>
Contribution to change of a ratio with 3 terms
user16439
# Contribution to change of a ratio with 3 terms

(Asked by: username_0 on 2018-03-02)

I have a micro dataset of firms' financial liabilities for 15 years. I would like to plot the contributions to change in leverage, decomposing the single contribution of bank loans, bonds and equity variations.
Leverage is built as:
Leverage=(Loans+Bonds)/(Loans+Bonds+Equity)

Therefore contributions to growth are positive if financial debt (Bank Loans or Bonds) increases or if its sum with capital decreases and vice versa.
I would be able to decompose with logarithm a simple ratio a/b but I have no clue on how to do it with 3 components. Any idea? Thanks

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2018-03-04. Score: 4

Suppose we want to find the contributions of the variables to the given function:

$$f(x,y)= {x\over{y}}$$

To find the growth rates of the given variables first take the logarithm of the function $f(x,y)$:
$$ln(f(x,y))=ln\left({x\over{y}}\right)=ln(x)-ln(y)$$

And then take the partial derivative of the variable you would like to analyze:

$${\partial ln\left({x\over{y}}\right)\over{\partial x}}={\dot{x}\over{x}}$$

$${\partial ln \left({x \over{y}} \right)\over{\partial y}}=-{\dot{y}\over{y}}$$

Where $\dot{x}$ and $\dot{y}$ are the derivatives of $x$ and $y$ respectively

Thus in your case if you would like to find the contributions to growth to leverage you can simply do the same thing as above.

i.e.

$$\mathfrak{L}={{l+b}\over{l+b+e}}$$

Where:

$\mathfrak{L}=$ leverage

$l=$ loans

$b=$ bonds

$e=$ equity

Thus

$$ln \left( \mathfrak{L} \right)= ln(l+b) -ln(l+b+e)$$

you can thus find the contributions to growth by the variables involved by taking the partial derivative of the variable you wish to examine.

Hope this is helpful!!


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