This paper derives analytically the growth rate of the social cost of carbon (SSC) on an optimal balanced growth path. More specifically, the paper examines a deterministic Ramsey model of optimal economic growth with carbon emissions. In this model, restrictions on technology and preferences are imposed that guarantee optimal balanced growth, i.e., that guarantee an optimal path with constant and positive economic growth and a constant stock of carbon in the atmosphere. The paper exploits these restrictions to show that the growth rate of the SCC on the optimal balanced growth path is negative, provided the elasticity of marginal utility of consumption with respect to consumption is larger than or equal to one. There seems to be consensus in the literature that this latter requirement is fulfilled in reality.Paper submitted to the special issue The Social Cost of Carbon