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Can someone please verify whether or not my knowledge is accurate.

From what I understand, the intent of the envelope theorem is to make a shortcut from indirect utility to the expenditure function. That being said, what you can do once you have obtained your indirect utility, is derive it in terms of I (income) and this will give you the expenditure function (hicksian demand). Is this correct or am I totally off?

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The Envelope theorem is a general mathematics result says that you can differentiate a value function with respect to a variable without implicitly differentiating the maximum argument variable.

Example: $V(p_x,p_y,w)=U(x(p_x,p_y,w), y(p_x,p_y,w))+\lambda(p_x,p_y,w)[w-p_x-p_y]$

If I want to see how utility at its optimal level changes when wealth changes, I can simply differentiate this equation with respect to wealth explicitly:

$V_3=\lambda$ - so the marginal utility of wealth is simply the Lagrange multiplier.

As you can see I did not differentiate anything else, even though wealth does appear in U. This is simply because if you did do that, those terms would drop out =0 as a result of the FOC

The relationship between indirect utility and expenditure is from the 'duality' between the two. The reason why $V(p,e(p,u))=w$ stems from the fact that the optimal arguments for both will be equivalent when the value of utility achieved in the utility max problem is the same utility level that we set as $U=\bar u$ in the expenditure minimization problem.

If you do the FOC for both utility max and expenditure min you get the tangency conditions:

$p_x/p_y=MRS$ in both cases. So if the target level $\bar u$ is what we achieve in the utility max problem, then the marshallian and hicksian demands will be the same. Then the value function for the expenditure min is:

$e(p_x,p_y,\bar u)=p_xh_x+p_yh_y=p_xx+p_yy=w$ and since $\bar u=V$

$e(p_x,p_y,V)=w$

Similarly, $V(p_x,p_y,w)=U(x,y)=U(h_x,h_y)=\bar u$ and since $w=e$

$V(p_x,p_y,e)=u$

You can find a lot more by searching for 'utility duality'.

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