Interpretation of $\frac{\partial }{\partial p_1}Q_1(p_1, p_2)/\frac{\partial}{\partial p_2} Q_1(p_1, p_2)$

Question

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!

Answer 1

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 $Q_1$.