I will set up a naive static/short run model to examine the case (so this post may be a bit long - I will try to dispense with some algebraic steps). I will use convenient functional forms, which are nevertheless consistent with usual assumptions.
FIRMS
There are $i=1,...,n$ identical, price taking firms. In the short run they maximize the objective function
$$A\ln\ell_i - (1+s_f+\xi)w\ell_i\tag{1} $$
where $A$ includes any component of the production function that is fixed in the short run, $\ell_i$ is the amount of labor firm $i$ employs, $s_f$ is the Employer's Social Security Fees (SSF) as a percentage over the "mixed" wage $w$. $\xi$ is a possible change in this percentage, which I include from the start.
The concept of "mixed wage" is central in actual labor markets: in most cases bilateral or union negotiations over the wage are carried in terms of the "mixed wage", not in terms of the "take home" wage.
Profit maximizing behavior will lead to market labor demand
$$L^d= n\cdot\frac {A}{(1+s_f+\xi)w} \tag{2}$$
WORKERS
There are $j=1,...,m$ workers, who posses one unit of labor and perform static maximization of the quasilinear utility function
$$U = c + \gamma \ln(1-\ell_j)\;\; s.t\;\; c= (1-s_w+\psi)w\ell_j \tag{3}$$
i.e. there is no consumption-saving decision here. $s_w$ is the "Employee's SSF" and $\psi$ is a possible change of this percentage (a positive $\psi$ implies lowering of the percentage)
Utility maximization leads to
$$L^s = m\cdot \frac{(1-s_w+\psi)w - \gamma}{(1-s_w+\psi)w} \tag{4}$$
Assuming that the labor markets clears, we have
$$L^d = L^s \implies n\cdot\frac {A}{(1+s_f+\xi)w} = m\cdot \frac{(1-s_w+\psi)w - \gamma}{(1-s_w+\psi)w}$$
$$\implies (nA/m)\frac {(1-s_w+\psi)}{(1+s_f+\xi)} = (1-s_w+\psi)w - \gamma$$
$$\implies w^* = \frac {(nA/m)}{(1+s_f+\xi)} + \frac {\gamma}{(1-s_w+\psi)} \tag{5}$$
Equation $(5)$ provides the first major conclusion :
If we increase "Employer's SSF" ($\xi >0$), the equilibrium mixed wage
will fall. But also, if we decrease "Employee's SSF" ($\psi >0$),
the equilibrium mixed wage will also fall.
This is because the "take-home wage" will increase for any given level of mixed wage, and so the labor supply curve will shift outwards in the $(w, L)$ space. Of course this result depends critically on labor-market clearing.
What will happen to individual worker's income?
Dividing $(2)$ by $m$ and using the equilibrium wage, equilibrium labor employed per worker will be
$$\ell_j^* = \frac {nA/m}{(1+s_f+\xi)w^*}$$ and so equilibrium take-home (disposable) labor income per worker will be
$$DI^*=(1-s_w+\psi)w^*\ell_j^* = (1-s_w+\psi)w^*\frac {nA/m}{(1+s_f+\xi)w^*} \tag{6}$$
$$\implies DI^* = \frac {1-s_w+\psi}{1+s_f+\xi }(nA/m) \tag{7}$$
Let's now start to implement's the adviser's idea. We start by the situation where $\xi=\psi = 0$. We want to determine $\xi$ and $\psi$ so that disposable income increases. This requires
$$DI^* \uparrow \implies \frac {1-s_w+\psi}{1+s_f+\xi } > \frac {1-s_w}{1+s_f}$$
$$\rightarrow DI^* \uparrow \implies \psi > \xi \frac {1-s_w}{1+s_f} \tag {8}$$
Since $(1-sw)/(1+s_f) <1 $ we conclude that
We do not need to decrease the Employee's SSF percentage as much as we
will increase the Employer's SSF percentage, in order to increase the
worker's disposable income. But the decrease should satisfy $(8)$.
But we want also to increase total Social Security Fees collected. Total Social Security Fees are
$$SSF^* = m\cdot\ell_j^*\cdot w^* \cdot (s_f+\xi + s_w - \psi) $$
$$\implies SSF^* = m\cdot \frac {nA/m}{(1+s_f+\xi)w^*} \cdot w^*\cdot (s_f+\xi + s_w - \psi) \tag{9}$$
$$\implies SSF^* = nA\cdot \frac {s_f+\xi + s_w - \psi}{(1+s_f+\xi)} $$
The condition to increase social security fees is
$$SSF^* \uparrow \implies \frac {s_f+\xi + s_w - \psi}{(1+s_f+\xi)} > \frac {s_f+s_w}{(1+s_f)}$$
$$\implies (1+s_f)(\xi-\psi) > \xi(s_f+s_w)$$
$$\implies \xi + s_f\xi - (1+s_f)\psi > s_f\xi + s_w\xi$$
$$\rightarrow SSF^* \uparrow \implies \psi < \xi \frac {1-s_w}{1+s_f} \tag{10}$$
But $(10)$ is the exact opposite condition than $(8)$. So:
There exists no combination of $\xi, \psi$ that will increase worker's
disposable income and increase total social security collections.
In other words, the adviser's proposal is infeasible.
Of course, I do not claim that this result generalizes to all models -neither do I have at this point a clear view of what are the crucial assumptions on which this infeasibility results rests.
next to the red triangle, do you mean to the left or right of the red? I don't know what is meant bythe transparent rectangle next to the red triangle. $\endgroup$