Rapid deployment of distributed energy resources (DERs), such as solar photovoltaics (PV), poses a risk to the distribution grid under high penetration. Therefore, studying hosting capacity (HC) limits considering grid physics and demand variability is crucial. This paper introduces an improved framework for determining the HC of radial distribution networks by enhancing an existing convex inner approximation (CIA) approach. The proposed method achieves a more accurate and larger inner approximation, resulting in better HC limits. We also consider time-varying demand and the design of objective functions to ensure equitable access to grid resources. A case study with solar PV integration is conducted using a modified IEEE-37 radial network to examine the impact of increased PV capacity, demonstrating that with no more than 5% annual solar PV energy curtailed, it is possible to increase solar PV hosting capacity by at least 50% with no negative grid impacts and a net positive economic impact when accounted for the cost of carbon. Results show that fair allocation methods can lead to higher net profits and reduced PV curtailment and CO2.