jmc

Multiply both sides of the first equation by $y$ and both sides of the second equation by $z$ to obtain $$\begin{array}{rl}xy+1& =y\\ yz+1& =z.\end{array}$$ Substituting $xy+1$ for $y$ in the second equation, we find $$(xy+1)z+1=z,$$ which simplifies to $$xyz+z+1=z.$$ Subtracting $z+1$ from both sides, we find that $xyz=z-(z+1)=\boxed{-1}.$