<p>a) The convention, when differentiating a real valued function <span class="math-container">$v$</span> wrt to a column vector <span class="math-container">$p$</span> of dimension <span class="math-container">$(G \times 1)$</span>, is that:
<span class="math-container">$$
\nabla _{p}v\left( p,w\right) \equiv \frac{\partial v}{\partial p}\left(
p,w\right) =\left( 
\begin{array}{c}
\frac{\partial v}{\partial p_{1}}\left( p,w\right)  \\ 
\vdots  \\ 
\frac{\partial v}{\partial p_{G}}\left( p,w\right) 
\end{array}%
\right). 
$$</span></p>

<p>b) When expenditures <span class="math-container">$w$</span> are not constant, the column vector <span class="math-container">$p$</span> with respect to which we take the derivative occurs twice in <span class="math-container">$v(p, e(p,\overline{u}))$</span>, and the chain rule has to be applied in order to obtain the total effect of the price change on the indirect utility: the first partial effect on utility is due to the price changes, and the second effect is through the adjustment in expenditures implied by the price change. Hence the expression:
<span class="math-container">\begin{eqnarray*}
&amp;&amp;\frac{\partial v}{\partial p}\left( p,e\left( p,u\right) \right) +\frac{%
\partial v}{\partial w}\left( p,e\left( p,u\right) \right) \frac{\partial e}{%
\partial p}\left( p,u\right)  \\
&amp;=&amp;\left( 
\begin{array}{c}
\frac{\partial v}{\partial p_{1}}\left( p,e(p,u)\right)  \\ 
\vdots  \\ 
\frac{\partial v}{\partial p_{G}}\left( p,e(p,u)\right) 
\end{array}%
\right) +\frac{\partial v}{\partial w}\left( p,e(p,u)\right) \left( 
\begin{array}{c}
\frac{\partial e}{\partial p_{1}}\left( p,u\right)  \\ 
\vdots  \\ 
\frac{\partial e}{\partial p_{G}}\left( p,u\right) 
\end{array}%
\right). 
\end{eqnarray*}</span></p>

CC BY-SA 4.0
2019-10-06T14:03:57.530Z
2019-10-06T14:03:57.530Z
Bertrand
<p>I am struggling a bit with the math in my first graduate microeconomics course. I'm not sure if this belongs here. If it doesn't, please direct me to a more appropriate place.</p>

<p>Below is one question along the with provided solution. The red part is the part I don't follow.</p>

<p><a href="https://i.sstatic.net/hehkJ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/hehkJ.png" alt="Solution image"></a></p>

<p>I don't understand how the derivative of <span class="math-container">$v(p, e(p,\overline{u}))$</span> w.r.t. the price vector can evaluate to the LHS of the given equation. </p>

<p>Perhaps I'm confused by the vector notation, so any step explaining this equation would be very much appreciated.</p>

<microeconomics><consumer-theory>
Prove that $h(p,u) = \nabla_p e(p,u)$ is implied by Roy's identity
avs
# Prove that $h(p,u) = \nabla_p e(p,u)$ is implied by Roy's identity

(Asked by: username_0 on 2019-10-06)

I am struggling a bit with the math in my first graduate microeconomics course. I'm not sure if this belongs here. If it doesn't, please direct me to a more appropriate place.

Below is one question along the with provided solution. The red part is the part I don't follow.

[![Solution image](https://i.sstatic.net/hehkJ.png)](https://i.sstatic.net/hehkJ.png)

I don't understand how the derivative of $v(p, e(p,\overline{u}))$ w.r.t. the price vector can evaluate to the LHS of the given equation.

Perhaps I'm confused by the vector notation, so any step explaining this equation would be very much appreciated.

**Tags:** microeconomics, consumer-theory


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## 1 answers

### Answer by username_1 on 2019-10-06. Score: 4

a) The convention, when differentiating a real valued function $v$ wrt to a column vector $p$ of dimension $(G \times 1)$, is that:
$$
\nabla \_{p}v\left( p,w\right) \equiv \frac{\partial v}{\partial p}\left(
p,w\right) =\left(
\begin{array}{c}
\frac{\partial v}{\partial p\_{1}}\left( p,w\right) \\
\vdots \\
\frac{\partial v}{\partial p\_{G}}\left( p,w\right)
\end{array}%
\right).
$$

b) When expenditures $w$ are not constant, the column vector $p$ with respect to which we take the derivative occurs twice in $v(p, e(p,\overline{u}))$, and the chain rule has to be applied in order to obtain the total effect of the price change on the indirect utility: the first partial effect on utility is due to the price changes, and the second effect is through the adjustment in expenditures implied by the price change. Hence the expression:
\begin{eqnarray\*}
&&\frac{\partial v}{\partial p}\left( p,e\left( p,u\right) \right) +\frac{%
\partial v}{\partial w}\left( p,e\left( p,u\right) \right) \frac{\partial e}{%
\partial p}\left( p,u\right) \\
&=&\left(
\begin{array}{c}
\frac{\partial v}{\partial p\_{1}}\left( p,e(p,u)\right) \\
\vdots \\
\frac{\partial v}{\partial p\_{G}}\left( p,e(p,u)\right)
\end{array}%
\right) +\frac{\partial v}{\partial w}\left( p,e(p,u)\right) \left(
\begin{array}{c}
\frac{\partial e}{\partial p\_{1}}\left( p,u\right) \\
\vdots \\
\frac{\partial e}{\partial p\_{G}}\left( p,u\right)
\end{array}%
\right).
\end{eqnarray\*}


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