The exposure of business applications to the web has considerably increased the variability of its workload patterns and volumes as the number of users/customers often grows and shrinks at various rates and times. Such application characteristics have increasingly demanded the need for flexible yet inexpensive computing infrastructure to accommodate variable workloads. The on-demand and per-use cloud computing model, specifically that of public Cloud Infrastructure Service Offerings (CISOs), has quickly evolved and adopted by majority of hardware and software computing companies with the promise of provisioning utility-like computing resources at massive economies of scale. However, deploying business applications on public cloud infrastructure does not lead to achieving desired economics and elasticity gains, and some challenges block the way for realizing its real benefits. These challenges are due to multiple differences between CISOs and application's requirements and characteristics. This article introduces a detailed analysis and discussion of the economics and elasticity challenges of business applications to be deployed and operate on public cloud infrastructure. This includes analysis of various aspects of public CISOs, modeling and measuring CISOs' economics and elasticity, application workload patterns and its impact on achieving elasticity and economics, economics-driven elasticity decisions and policies, and SLA-driven monitoring and elasticity of cloud-based business applications. The analysis and discussion are supported with motivating scenarios for cloud-based business applications. The paper provides a multi-lenses overview that can help cloud consumers and potential business application's owners to understand, analyze, and evaluate important economics and elasticity capabilities of different CISOs and its suitability for meeting their business application's requirements.