  In this paper we consider two problems on optimal implementation delay of
taxation with trade-off for spectrally negative L\'{e}vy insurance risk
processes. In the first case, we assume that an insurance company starts to pay
tax when its surplus reaches a certain level $b$ and at the termination time of
the business there is a terminal value incurred to the company. The total
expected discounted value of tax payments plus the terminal value is maximized
to obtain the optimal implementation level $b^*$. In the second case, the
company still pays tax subject to an implementation level $a$ but with capital
injections to prevent bankruptcy. The total expected discounted value of tax
payments minus the capital injection costs is maximized to obtain the optimal
implementation level $a^*$. Numerical examples are also given to illustrate the
main results in this paper.
