We develop an overlapping generations endogenous growth model with stocks of produced capital, human capital, a non-renewable resource, and irreversibly accumulated greenhouse gases in deterministic and stochastic versions. The model allows for analyzing different elasticities of substitution. We present a full analytical solution and characterization of the transition dynamics. We show that, as a rule of thumb, the social cost of carbon grow at a rate equal to the economy's growth rate divided by the elasticity of substitution. We analytically study sensitivity of the social cost of carbon with respect to key parameters: the intergenerational discount rate, the elasticity of substitution, and climate uncertainty. We show that the social cost of carbon explode at a finite level of log-normally distributed climate uncertainty. We illustrate results in a calibrated version of the model.