  We consider the directed first passage percolation model on ${\bf Z}^2$. In
this model, we assign independently to each edge $e$ a passage time $t(e)$ with
a common distribution $F$. We denote by $\vec{T}({\bf 0}, (r,\theta))$ the
passage time from the origin to $(r, \theta)$ by a northeast path for $(r,
\theta)\in {\bf R}^+\times [0,\pi/2]$. It is known that $\vec{T}({\bf 0}, (r,
\theta))/r$ converges to a time constant $\vec{\mu}_F (\theta)$. Let
$\vec{p}_c$ denote the critical probability for oriented percolation. In this
paper, we show that the time constant has a phase transition divided by
$\vec{p}_c$, as follows:
  (1) If $F(0) < \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ for all $0\leq
\theta\leq \pi/2$.
  (2) If $F(0) = \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ if and only if
$\theta\neq \pi/4$.
  (3) If $F(0)=p > \vec{p}_c$, then there exists a percolation cone between
$\theta_p^-$ and $\theta_p^+$ for
  $0\leq \theta^-_p< \theta^+_p \leq \pi/2$ such that $\vec{\mu} (\theta) >0$
if and only if $\theta\not\in [\theta_p^-, \theta^+_p]$. Furthermore, all the
moments of $\vec{T}({\bf 0}, (r, \theta))$ converge whenever $\theta\in
[\theta_p^-, \theta^+_p]$. As applications, we describe the shape of the
directed growth model on the distribution of $F$. We give a phase transition
for the shape divided by $\vec{p}_c$.

Probability