Title: Regulation in a mean-field investment game with climate damage.

Abstract: We develop a mean-field model of firms investing in carbon-intensive (brown) capital, where productivity declines due to temperature-related climate damages linked to cumulative emissions. Firms aim to maximise expected discounted profits under stochastic capital dynamics driven by idiosyncratic shocks, facing a trade-off between investment-driven growth and increased emissions. We characterise the unique mean-field equilibrium as the solution to a coupled system of forward-backward ordinary differential equations. This framework also enables the analysis of optimal regulatory instruments — such as carbon emission taxes and brown production phase-out timelines — in line with long-term climate goals.