  We introduce a highly stylized, yet non trivial model of the economy, with a
public and private sector coupled through a wealth tax and a redistribution
policy. The model can be fully solved analytically, and allows one to address
the question of optimal taxation and of wealth inequalities. We find that
according to the assumption made on the relative performance of public and
private sectors, three situations are possible. Not surprisingly, the optimal
wealth tax rate is either 0% for a deeply dysfunctional government and/or
highly productive private sector, or 100 % for a highly efficient public sector
and/or debilitated/risk averse private investors. If the gap between the
public/private performance is moderate, there is an optimal positive wealth tax
rate maximizing economic growth, even -- counter-intuitively -- when the
private sector generates more growth. The compromise between profitable private
investments and taxation however leads to a residual level of inequalities. The
mechanism leading to an optimal growth rate is related the well-known
explore/exploit trade-off.
