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

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.


1 Answer 1


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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