Use Roy's Theorem to prove that ...

Question

I have an advanced microeconomics theory related question:

Use Roy's theorem to prove that $s_i(p,y)= -\frac{\partial v(p,y)}{\partial lnp_i}/\frac{\partial v(p,y)}{\partial lny}$.

This question has been bothering me for quite sometime now and so I decided to ask for some help here.

So, I'm assuming $s_i$ is budget share of good $i$ and $p_i$ is the price of good $i$. What is $v(p,y)$? Is this an indirect utility function? If so, is $y$ income? — jmbejara, Nov 05 '13 at 15:23
You could give us the answer and accept it, so this question is designated as answered. ;) — Nameless, Nov 07 '13 at 12:32

score 0 · Accepted Answer · 2013-11-07T23:01:10.867

Suppose $s_i$ is the budget share for good $i$, so $s_i = \frac{x_{i}p_{i}}{y}$.

Since $\frac{\partial v (p,y)}{\partial lnp_i} = p_i\frac{\partial v}{\partial p_i}$ and $\frac{\partial v (p,y)}{\partial lny} = y\frac{\partial v}{\partial y}$.

So $-\frac{\frac{\partial v (p,y)}{\partial lnp_i}}{\frac{\partial v (p,y)}{\partial lny}} = \frac{-p_i}{y} \frac{\frac{\partial v}{\partial p_i}}{\frac{\partial v}{\partial y}} = \frac{-p_i}{y} x_i^{*}(p,y)$ (by Roy's Theorem).

Hence $\frac{-p_i}{y} x_i^{*}(p,y)= s_i(p,y)$.

Use Roy's Theorem to prove that ...

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