Econ Intuition for Jacobian inverse in demand system

Question

Consider the following simple linear demand system (in vector notation) with n different products

Demand: $\quad\mathbf{q=B\left(a-p\right)}$

Inverse demand: $\quad\mathbf{p=a-B^{-1}q}$

where $\mathbf{p}$ is the vector of prices and $\mathbf{q}$ is the vector of quantities supplied. The Jacobians of these two equations

$\dfrac{\partial\mathbf{q}}{\partial\mathbf{p}}=-\mathbf{B}\quad$ and $\quad\dfrac{\partial\mathbf{p}}{\partial\mathbf{q}}=-\mathbf{B}^{-1}$

tell you how prices react to a change in quantity supplied and vice-versa. Now, consider the example:

$\mathbf{B}=\left[\begin{array}{cc} \frac{4}{3} & -\frac{2}{3}\\ -\frac{2}{3} & \frac{4}{3} \end{array}\right]\quad$ so that $\quad\mathbf{B}^{-1}=\left[\begin{array}{cc} 1 & .5\\ .5 & 1 \end{array}\right]$

then it is easy to see that, for some product $i$:

$\dfrac{\partial q_{i}}{\partial p_{i}}\neq\left(\dfrac{\partial p_{i}}{\partial q_{i}}\right)^{-1}$

I understand that that is true mathematically and will generally be the case, but what does it mean in terms of economic intuition? Why does the demand curve for product $i$ slope differently depending on whether I put $q_{i}$ on the vertical axis rather than the horizontal axis?

Answer 1

For the 2x2 case being considered, write $$\mathbf{B}=\left[\begin{array}{cc} b_{1,1} & b_{1,2}\\ b_{2,1} & b_{2,2} \end{array}\right].\quad$$

It follows that the element (1,1) in $B^{-1}$ is given by $\frac{b_{2,2}}{b_{1,1}b_{2,2}-b_{1,2}b_{2,1}}$. Notice that $$\frac{\partial q_1(p_1,p_2)}{\partial p_1}=(\frac{\partial p_1(q_1,q_2)}{\partial q_1 })^{-1}$$ implies $$b_{1,1}=(\frac{b_{2,2}}{b_{1,1}b_{2,2}-b_{1,2}b_{2,1}})^{-1},$$ which can only occur if either $b_{1,2}=0$, $b_{2,1}=0,$ or both.

To better understand the intuition for this result, consider the demand function $q_1(p_1,p_2).$ Suppose we are interested in the demand curve for good 1 when $p_2=\tilde{p_2}$, which is a plot of the function $q_1(p_1,\tilde{p_2})$ with $p_1$ on the vertical axis and $q_1$ on the horizontal axis. The slope of this demand curve is given by $(\frac{\partial q_1(p_1,p_2)}{\partial p_1})^{-1}=-\frac{1}{b_{1,1}}$.

Now consider the inverse demand for good 1, which gives us maximum marginal willingness to pay for good 1 conditional on quantities of good 1 and good 2. If we want to plot this demand curve we must fix the quantity of good 2, say at $x_2=\tilde{x_2}.$ Suppose we want this demand curve to correspond to the demand curve we plotted for $q_1(p_1,\tilde{p_2})$, meaning $\tilde{x_2}$ must be the quantity demanded at $\tilde{p_2}$. However, to find the quantity demanded of good 2 at $\tilde{p_2}$,we must condition on the price of good 1. Likewise, to find the marginal willingness to pay of good 2 at quantity $\tilde{x_2}$, we must condition on the quantity of good 1. Both of the previous statements make no sense, as we are trying to plot the relationship between $x_1$ and $p_1$, so we cannot fix either of them.

The basic intuition is that when we invert $B$ to solve for inverse demands, we are accounting for the effects of cross-price effects in writing our marginal willingness to pay (inverse demand) functions. In economics terms, the following two expressions are not equal unless cross-price effects are zero:

1. Marginal effect of the quantity of good 1 on willingness-to-pay for good 1, conditional on the quantity of good 2.
2. Inverse of the marginal effect of the price of good 1 on quantity demanded of good 1, conditional on the price of good 2.