<p>For the 2x2 case being considered, write
<span class="math-container">$$\mathbf{B}=\left[\begin{array}{cc}
b_{1,1} &amp; b_{1,2}\\
b_{2,1} &amp; b_{2,2}
\end{array}\right].\quad$$</span></p>

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

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

<p>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 <span class="math-container">$x_2=\tilde{x_2}.$</span> Suppose we want this demand curve to correspond to the demand curve we plotted for <span class="math-container">$q_1(p_1,\tilde{p_2})$</span>, meaning <span class="math-container">$\tilde{x_2}$</span> must be the quantity demanded at <span class="math-container">$\tilde{p_2}$</span>. However, to find the quantity demanded of good 2 at <span class="math-container">$\tilde{p_2}$</span>,we must condition on the price of good 1. Likewise, to find the marginal willingness to pay of good 2 at quantity <span class="math-container">$\tilde{x_2}$</span>, 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 <span class="math-container">$x_1$</span> and <span class="math-container">$p_1$</span>, so we cannot fix either of them. </p>

<p>The basic intuition is that when we invert <span class="math-container">$B$</span> 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:</p>

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

CC BY-SA 4.0
2019-03-16T01:50:10.700Z
Thanks a lot. I like your explanation. Cheers
bbecon
CC BY-SA 4.0
2019-03-16T01:53:33.620Z
Very interesting question. Thanks for posting.
dlnB
CC BY-SA 4.0
2019-03-15T03:11:48.523Z
2019-03-16T01:51:27.603Z
2019-03-16T01:51:27.603Z
dlnB
<p>Consider the following simple linear demand system (in vector notation) with n different products</p>

<p>Demand: <span class="math-container">$\quad\mathbf{q=B\left(a-p\right)}$</span></p>

<p>Inverse demand: <span class="math-container">$\quad\mathbf{p=a-B^{-1}q}$</span></p>

<p>where <span class="math-container">$\mathbf{p}$</span> is the vector of prices and <span class="math-container">$\mathbf{q}$</span> is the vector of quantities supplied. The Jacobians of these two equations</p>

<p><span class="math-container">$\dfrac{\partial\mathbf{q}}{\partial\mathbf{p}}=-\mathbf{B}\quad$</span>
  and  <span class="math-container">$\quad\dfrac{\partial\mathbf{p}}{\partial\mathbf{q}}=-\mathbf{B}^{-1}$</span></p>

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

<p><span class="math-container">$\mathbf{B}=\left[\begin{array}{cc}
\frac{4}{3} &amp; -\frac{2}{3}\\
-\frac{2}{3} &amp; \frac{4}{3}
\end{array}\right]\quad$</span> so that <span class="math-container">$\quad\mathbf{B}^{-1}=\left[\begin{array}{cc}
1 &amp; .5\\
.5 &amp; 1
\end{array}\right]$</span></p>

<p>then it is easy to see that, for some product <span class="math-container">$i$</span>:</p>

<p><span class="math-container">$\dfrac{\partial q_{i}}{\partial p_{i}}\neq\left(\dfrac{\partial p_{i}}{\partial q_{i}}\right)^{-1}$</span></p>

<p>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 <span class="math-container">$i$</span>  slope differently depending on whether I put <span class="math-container">$q_{i}$</span> on the vertical axis rather than the horizontal axis?</p>

<elasticity-of-substitution><linear-algebra><demand>
Econ Intuition for Jacobian inverse in demand system
bbecon
# Econ Intuition for Jacobian inverse in demand system

(Asked by: username_0 on 2019-03-15)

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?

**Tags:** elasticity-of-substitution, linear-algebra, demand

---

## 1 answers

### Answer by username_1 on 2019-03-15. Score: 7 | Accepted answer

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.

> **Comments:**
> **username_0**: Thanks a lot. I like your explanation. Cheers  
> **username_1**: Very interesting question. Thanks for posting.  

---

