<p>One interpretation I can offer. The demand function can be expressed as:</p>

<p><span class="math-container">$$Q_1 = Q_1(p_1,p_2)$$</span></p>

<p>Let us take the total differential:</p>

<p><span class="math-container">$$dQ_1 = \frac{\partial Q_1(p_1,p_2)}{\partial p_1}dp_1+\frac{\partial Q_1(p_1,p_2)}{\partial p_2}dp_2$$</span></p>

<p>Assume that <span class="math-container">$Q_1$</span> remains unchanged with respect to a change in prices. This implies that <span class="math-container">$dQ_1=0$</span>. Solving the equation:</p>

<p><span class="math-container">$$\frac{d p_2}{d p_1} = - \frac{\frac{\partial Q_1(p_1,p_2)}{\partial p_1}}{\frac{\partial Q_1(p_1,p_2)}{\partial p_2}}$$</span></p>

<p>The above expression implies the following (since both partial derivatives are negative):</p>

<p><span class="math-container">$$t = \left|\frac{dp_2}{dp_1}\right| = \frac{\frac{\partial Q_1(p_1,p_2)}{\partial p_1}}{\frac{\partial Q_1(p_1,p_2)}{\partial p_2}}$$</span></p>

<p><span class="math-container">$t$</span> can be interpreted as follows: if good 1 became more expensive, how much would the price of good 2 need to change (decrease) such that the demand for good 1 remains unchanged. </p>

<p>Mathematically, <span class="math-container">$t$</span> represents the magnitude of the slope at (<span class="math-container">$p_1,p_2$</span>) on the level curve when we fix <span class="math-container">$Q_1$</span>.</p>

CC BY-SA 4.0
2019-12-03T20:34:55.627Z
2019-12-03T20:34:55.627Z
lunar_props
<p>I am interested in an economic interpretation for the ratio of partial derivatives of a demand function <span class="math-container">$Q_1(p_1, p_2)$</span>, which is
<span class="math-container">\begin{equation}
t=\frac{\frac{\partial}{\partial p_1}Q_1(p_1, p_2)}{\frac{\partial}{\partial p_2}Q_1(p_1, p_2)}.
\end{equation}</span>
Assume the goods are complements so that both partial derivatives are negative. </p>

<p>This has the same form as the MRS in consumer theory, but the intuition does not readily apply. </p>

<p>I have been thinking about this as the ratio of demand sensitivities: if <span class="math-container">$t&gt;1$</span> then the demand for good 1 is more sensitive to changes in <span class="math-container">$p_1$</span>; if <span class="math-container">$t&lt;1$</span>, then the demand for good 1 is more sensitive to changes in <span class="math-container">$p_2$</span>.</p>

<p>I would also say this is somewhat related to the cross price elasticity of demand, but the formula is not the same. </p>

<p>Any interpretations or intuitions would be great!</p>

<microeconomics><mathematical-economics><supply-and-demand><industrial-organisation>
Interpretation of $\frac{\partial }{\partial p_1}Q_1(p_1, p_2)/\frac{\partial}{\partial p_2} Q_1(p_1, p_2)$
Hal_Incandenza
# Interpretation of $\frac{\partial }{\partial p_1}Q_1(p_1, p_2)/\frac{\partial}{\partial p_2} Q_1(p_1, p_2)$

(Asked by: username_0 on 2019-12-03)

I am interested in an economic interpretation for the ratio of partial derivatives of a demand function $Q\_1(p\_1, p\_2)$, which is
\begin{equation}
t=\frac{\frac{\partial}{\partial p\_1}Q\_1(p\_1, p\_2)}{\frac{\partial}{\partial p\_2}Q\_1(p\_1, p\_2)}.
\end{equation}
Assume the goods are complements so that both partial derivatives are negative.

This has the same form as the MRS in consumer theory, but the intuition does not readily apply.

I have been thinking about this as the ratio of demand sensitivities: if $t>1$ then the demand for good 1 is more sensitive to changes in $p\_1$; if $t<1$, then the demand for good 1 is more sensitive to changes in $p\_2$.

I would also say this is somewhat related to the cross price elasticity of demand, but the formula is not the same.

Any interpretations or intuitions would be great!

**Tags:** microeconomics, mathematical-economics, supply-and-demand, industrial-organisation

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## 1 answers

### Answer by username_1 on 2019-12-03. Score: 7

One interpretation I can offer. The demand function can be expressed as:

$$Q\_1 = Q\_1(p\_1,p\_2)$$

Let us take the total differential:

$$dQ\_1 = \frac{\partial Q\_1(p\_1,p\_2)}{\partial p\_1}dp\_1+\frac{\partial Q\_1(p\_1,p\_2)}{\partial p\_2}dp\_2$$

Assume that $Q\_1$ remains unchanged with respect to a change in prices. This implies that $dQ\_1=0$. Solving the equation:

$$\frac{d p\_2}{d p\_1} = - \frac{\frac{\partial Q\_1(p\_1,p\_2)}{\partial p\_1}}{\frac{\partial Q\_1(p\_1,p\_2)}{\partial p\_2}}$$

The above expression implies the following (since both partial derivatives are negative):

$$t = \left|\frac{dp\_2}{dp\_1}\right| = \frac{\frac{\partial Q\_1(p\_1,p\_2)}{\partial p\_1}}{\frac{\partial Q\_1(p\_1,p\_2)}{\partial p\_2}}$$

$t$ can be interpreted as follows: if good 1 became more expensive, how much would the price of good 2 need to change (decrease) such that the demand for good 1 remains unchanged.

Mathematically, $t$ represents the magnitude of the slope at ($p\_1,p\_2$) on the level curve when we fix $Q\_1$.


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