# Envelope Theorem in Hopkins and Kornienko (2010)

This is from Hopkins and Kornienko (2010). In this model, $$x$$ is investments, $$s$$ is status, and $$y=z-x$$ is leisure, where $$z$$ is endowments. $$x(r)$$ is the optimal investment, and the relative investments determine the status in the model. Here they characterize the agent with the rank, $$r$$, and derive the reduced form utility. But, I am not quite sure about how the envelope theorem works here.

If I differentiate $$U(r)$$ with respect to $$r$$ and suppress the arguments, I have that $$U_x x' + U_y(y'-x') + U_s s'.$$ I know that $$Z'(r) = \frac{1}{g(Z(r))}$$, where $$g$$ is the density of $$z$$. This implies that the first term and third term and $$U_yx'$$ turn out to be zero to make the above equation equal to Equation (8). My understanding is that $$U_x = U_s = 0$$ since $$x(r)$$ is the optimal investment. But, I don't know how to explain the elimination of $$U_yx'$$. Can anyone give me some help? • What does $p$ stand for in $Z'(p)$? – Bertrand May 12 '19 at 18:34
• @Bertrand It should be $Z'(r)$. Sorry. – shk910 May 13 '19 at 0:05

$$U_x(x,Z(r)-x,S(r))x'(r) - U_y(x,Z(r)-x,S(r))x'(r) + U_s(x,Z(r)-x,S(r))S'(r)=0.$$
When differentiating $${\bf U}(r)$$ with respect to $$r$$, we find your expression: $$U_x(x(r),Z(r)-x(r),S(r))x'(r) + U_y(x(r),Z(r)-x(r),S(r))(Z'(r)-x'(r)) + U_s(x(r),Z(r)-x(r),S(r))S'(r).$$
Replacing the first equation into the last one yields $${\bf U}'(r)=U_y(x(r),Z(r)-x(r),S(r))Z'(r).$$
Now, $$Z'(r)=1/g(Z(r))$$ which gives equation (8).