The increasing prominence of Socially Responsible Investing (SRI) has led institutional investors to incorporate ESG ratings into their decision-making, yet the impact of these ratings on firm outcomes remains unclear. This paper examines how MSCI ESG labels influence firms' ownership by ESG institutional investors and their perceived cost of capital. I employ a Regression Discontinuity (RD) design around the two cutoffs where MSCI's numerical ESG scores are converted into distinct ESG labels. I find that around the Top-Cutoff, i.e., among high-ESG firms, those with better ESG labels have higher ownership by ESG institutional investors and have lower perceived cost of capital. Conversely, around the Bottom-Cutoff, i.e., among low-ESG firms, those with worse ESG labels show a similar pattern. I argue that higher ownership by ESG institutional investors is the reason behind the lower perceived cost of capital at both cutoffs. Moreover, I examine monetary and non-monetary incentives for the opposite behavior of ESG institutional investors around these cutoffs. These findings contribute to the SRI literature by demonstrating how ESG labels affect ESG institutional investors' portfolio decisions and firms' cost of capital, revealing the nuanced strategies that ESG investors use based on the firms' ESG performance.