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?