  We consider a one-period Kyle (1985) framework where the insider can be
subject to a penalty if she trades. We establish existence and uniqueness of
equilibrium for virtually any penalty function when noise is uniform. In
equilibrium, the demand of the insider and the price functions are in general
non-linear and remain analytically tractable because the expected price
function is linear. We use this result to investigate the trade off between
price efficiency and 'fairness': we consider a regulator that wants to minimise
post-trade standard deviation for a given level of uninformed traders' losses.
The minimisation is over the function space of penalties; for each possible
penalty, our existence and uniqueness theorem allows to define unambiguously
the post-trade standard deviation and the uninformed traders' losses that
prevail in equilibrium.Optimal penalties are characterized in closed-form. They
must increase quickly with the magnitude of the insider's order for small
orders and become flat for large orders: in cases where the fundamental
realizes at very high or very low values, the insider finds it optimal to trade
despite the high penalty. Although such trades-if they occur-are costly for
liquidity traders, they signal extreme events and therefore incorporate a lot
of information into prices. We generalize this result in two directions by
imposing a budget constraint on the regulator and considering the cases of
either non-pecuniary or pecuniary penalties. In the first case, we establish
that optimal penalties are a subset of the previously optimal penalties: the
patterns of equilibrium trade volumes and prices is unchanged. In the second
case, we also fully characterize the constrained efficient points and penalties
and show that new patterns emerge in the demand schedules of the insider trader
and the associated price functions.
