With the development of integrated electricity and gas markets (IEGM), the participation of multi-energy demands (MEDs) is an effective measure to achieve a win-win goal of market efficiency improvement and demand cost reduction. The MEDs, composed of terminal users and energy conversion equipment (ECE), can adjust purchased electricity and gas with the variation in energy prices. Such adjustment behaviors of purchased energy in response to energy prices are defined as the multi-elasticity of MEDs. Considering that, this paper aims to analyze and quantify the multi-elasticity of MEDs and exploit it using a market-based approach. Firstly, the volume-price bidding curves are proposed to describe the multi-elasticity of MEDs considering the price-response features of terminal loads and multi-energy substitution features of ECE. Based on the proposed bidding curves, the energy consumption behaviors of MEDs with the change in prices can be effectively quantified. Moreover, a decentralized two-stage market framework is proposed to exploit the multi-elasticity in the IEGM clearing based on the multi-elasticity bidding curves of MEDs. In the proposed framework, the electricity market and gas market are cleared separately with the information exchange of the MEDs' clearing results of purchased gas and electricity. Furthermore, the consensus ADMM-based algorithm is applied to coordinately solve the IEGM clearing problem integrated with the multi-elasticity of MEDs. Numerical simulations are carried out for different test systems to validate the effectiveness of the proposed models.