Envelope Theorem in Keen and Slemrod (2017) This question pertains to the paper "Optimal Tax Administration" by Slemrod and Keen (2017). The IMF working paper is freely available on SSRN, however, it is not necessary to know the paper in order to understand the question.

I am confused on how to arrive at equation (5) in the working paper version.

Suppose there is a welfare function:

$W(t,\alpha)= wl - tz(t,\alpha) -c(e,\alpha) - \psi(l) + v(tz(t,\alpha) -a(\alpha)) $ 

Here, $v$ is an increasing and concave function, $\psi$ is an incresing and convex function, $t$ is the tax rate, $\alpha$ is tax enforcement, $c$ is evasion/compliance cost and $z$ is declared income, which depends on $t$ and $\alpha$.

Declared income is given by $z=wl(t,w) - e(t,\alpha)$, where $w$ is the wage rate, $l$ is hours worked and $e$ is concealed income. Both $l$ and $e$ are derived optimally.

The first order conditions for $l$ and $e$ are given by:

$(1-t)w - \psi'(l) = 0$

$t - c_e(e,\alpha) = 0$

Here $c_e$ denotes the derivative of $c$ with respect to $e$ and $\psi'$ is the derivative of $\psi$.

I am interested in $\frac{dW}{dt}$.

The author's invoke the envelope property to arrive at:

$\frac{dW}{dt}= -z + v'*(z + tz_t)$

Here $z_t$ is the derivative of $z$ with resprect to t and $v'$ is the derivative of $v$. 

How  can I arrive at this expression?
At first glance, it appears the answer should be $\frac{dW}{dt}= -(z + tz_t) + v'*(z + tz_t)$, as $l$ and $e$ are chosen optimally and the envelope theorem cancels these terms out. Furthermore, one could expect both $z_t$ terms in the final derivative to drop out due to the envelope theorem. However, one and only one $z_t$ term remains in the authors' calculations.

The envelope theorem is just an application of first order conditions. In this case, it is better to simply use the first order conditions we know to derive the results, instead of using the theorem directly. I believe the authors invoke the term "envelope property" to concisely convey that they are using the previously found first order conditions. 

Taking the total derivative we arrive at: 

$\frac{dW}{dt}= wl_t -z - tz_t -c_ee' - \psi'l_t + v'*(z + tz_t)$

Inserting the expression for $z_t$ and grouping some terms we have:

$\frac{dW}{dt}= l_t[w(1-t)-\psi'] + e'[t-c_e] - z + v'*(z + tz_t)$

Note that the terms in square brackets are the same as the first order conditions for $l$ and $e$ and hence equal to zero. This leaves us with:

$\frac{dW}{dt}= - z + v'*(z + tz_t)$