Econ Intuition for Jacobian inverse in demand system

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?

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.
• Thanks a lot. I like your explanation. Cheers – bbecon Mar 16 '19 at 1:50
• Very interesting question. Thanks for posting. – dlnB Mar 16 '19 at 1:53