  We present a phase-plane analysis of cosmologies containing a barotropic
fluid with equation of state $p_\gamma = (\gamma-1) \rho_\gamma$, plus a scalar
field $\phi$ with an exponential potential $V \propto \exp(-\lambda \kappa
\phi)$ where $\kappa^2 = 8\pi G$. In addition to the well-known inflationary
solutions for $\lambda^2 < 2$, there exist scaling solutions when $\lambda^2 >
3\gamma$ in which the scalar field energy density tracks that of the barotropic
fluid (which for example might be radiation or dust). We show that the scaling
solutions are the unique late-time attractors whenever they exist. The
fluid-dominated solutions, where $V(\phi)/\rho_\gamma \to 0$ at late times, are
always unstable (except for the cosmological constant case $\gamma = 0$). The
relative energy density of the fluid and scalar field depends on the steepness
of the exponential potential, which is constrained by nucleosynthesis to
$\lambda^2 > 20$. We show that standard inflation models are unable to solve
this `relic density' problem.
