This paper presents a phase-plane analysis of cosmologies containing a barotropic fluid with equation of state $ p_\gamma = (\gamma - 1)\rho_\gamma $ and a scalar field with an exponential potential $ V \propto \exp(-\lambda \kappa \phi) $. The analysis shows that for $ \lambda^2 < 3\gamma $, the scalar field energy density dominates at late times, while for $ \lambda^2 > 3\gamma $, the scalar field energy density remains a fixed fraction of the total density. The paper also discusses the implications of these results for cosmological models, particularly the relic density problem. It is shown that standard inflation models are unable to solve this problem, as they do not significantly dilute the initial density of the exponential potential. The paper also discusses the role of inflation in addressing the relic density problem and concludes that only inflation models with effectively constant energy density and an exponentially large number of e-foldings can weaken the bound. The paper emphasizes that the results are based on the assumption that there is no direct coupling between the exponential potential and other matter, and that the only interaction is gravitational. The analysis shows that the scaling solution is the unique late-time attractor for $ \lambda^2 > 3\gamma $, and that the scalar field energy density remains constant due to the large friction term in the evolution equation. The paper also discusses the implications of these results for the early universe, particularly the role of scalar fields with exponential potentials in driving inflation. The paper concludes that the results have important implications for cosmological models and that further investigation is needed to fully understand the role of scalar fields with exponential potentials in the universe.This paper presents a phase-plane analysis of cosmologies containing a barotropic fluid with equation of state $ p_\gamma = (\gamma - 1)\rho_\gamma $ and a scalar field with an exponential potential $ V \propto \exp(-\lambda \kappa \phi) $. The analysis shows that for $ \lambda^2 < 3\gamma $, the scalar field energy density dominates at late times, while for $ \lambda^2 > 3\gamma $, the scalar field energy density remains a fixed fraction of the total density. The paper also discusses the implications of these results for cosmological models, particularly the relic density problem. It is shown that standard inflation models are unable to solve this problem, as they do not significantly dilute the initial density of the exponential potential. The paper also discusses the role of inflation in addressing the relic density problem and concludes that only inflation models with effectively constant energy density and an exponentially large number of e-foldings can weaken the bound. The paper emphasizes that the results are based on the assumption that there is no direct coupling between the exponential potential and other matter, and that the only interaction is gravitational. The analysis shows that the scaling solution is the unique late-time attractor for $ \lambda^2 > 3\gamma $, and that the scalar field energy density remains constant due to the large friction term in the evolution equation. The paper also discusses the implications of these results for the early universe, particularly the role of scalar fields with exponential potentials in driving inflation. The paper concludes that the results have important implications for cosmological models and that further investigation is needed to fully understand the role of scalar fields with exponential potentials in the universe.