# Envelope Theorem

Say I'm solving for consumption given income, and we assume that an interior solution exists:

$$\max_c U(c, Y)\\ C(Y) = \arg\max_c U(c, Y)\\$$

However, there is no closed-form solution for $C(Y)$. But I know it is cont. differentiable. Now I am not interested in $C(Y)$, but I want to know the sign of $C'(Y)$. Here was my approach:

Define the first-order condition as $G(c,Y) = 0$. Differentiate:

$$\frac{d}{d Y} G(c,Y) = \frac{\partial G(c,Y)}{\partial Y} + \frac{\partial G(c,Y)}{\partial c}\frac{d c}{d Y} = 0$$

Now I am somewhat stuck: How can I actually compute $\frac{d c}{d Y}$? That's what I'm, after all. I seem to remember that the envelope theorem says that $\partial c/\partial Y$ should be zero. How do I proceed?

The first order condition is $U_1(C(Y),Y)=0$, where $U_i$ means the partial derivative with respect to the $i$th argument.

Now, totally differentiate with respect to $Y$: $$U_{12}(C(Y),Y)+U_{11}(C(Y),Y)C'(Y)=0.$$ This can be rearranged to give $$C'(Y)=-\frac{U_{12}(C(Y),Y)}{U_{11}(C(Y),Y)}.$$

Usually, we expect the objective function is concave (i.e. $U_{11}<0$), so the sign of $C'(Y)$ is the same as the sign of $U_{12}(C(Y),Y)$.

Usually, when applying this kind of comparative static trick, the theory tells us what the sign of $U_{12}$ should be. As an example, if a entrepreneur's output as a function of effort and skill is $y(e,s)$, you might think that $y_{12}>0$ (i.e. an increase in effort yields a bigger increase in output for more skilled people). The analagous exercise to the one you are conducting above would then yield $y_{12}>0\implies e'(s)>0$ so more skilled entrepreneurs try harder.

It's hard to know what the appropriate assumption on $U_{12}$ in your example might be because it's not clear why income is in the utility function in the first place and why $U$ isn't just increasing in $c$. Perhaps you have some application in mind that can pin this down.

The envelope theorem would have a different use here. Suppose we found the optimal $C(Y)$ and are interested in the question "evaluated at $C(Y)$, how does an increase in income affect utility"?

$$\frac{d U(C(Y),Y)}{dY}=\underbrace{U_1(C(Y),Y)}_{=0}C'(Y)+U_2(C(Y),Y)$$
The envelope theorem tells us that we can ignore the first term (i.e. the indirect effect of $Y$ on $C$) because the first order condition for optimal choice of $C$ (i.e. $U_1=0$) guarantees this effect is zero.