For many environmental problems regulators are uncertain about both the costs of abatement and the stock-related damages. For example, governments have imperfect information about damages caused by greenhouse gas stocks, and also about the costs of abating greenhouse gas emissions. They acquire information over time, and this learning affects future expected payoffs. The ability to learn, either about abatement costs or about environmental damages, influence current decisions. Using numerical solutions of a simple model of global warming, we show how endogenous learning about abatement costs and environmental damages influence the choice of taxes or quotas. Much of the literature concerning uncertainty and learning about damages from stock externalities such as climate change assumes that the uncertainty will eventually be resolved. These papers focus on the effect of "passive" learning with which information arrives exogenously so that the mere passage of time reduces uncertainty, ignoring the possible impact of the regulator's optimal decisions on the learning process. Passive learning is inferior to "active" learning in cases where the regulator can influence the amount of new information. For example, stock externalities and environmental damages following emission decisions convey information about unknown damage parameters. We show how uncertainty about damages, together with anticipated active learning, influences optimal regulation and the ranking of policies (taxes versus quotas). We consider both parametric uncertainty and stochasticity in environmental damages. The parametric uncertainty arises because the regulator does not know the true value of some parameters in the damage function, e.g. the slope of the marginal damage (g). The stochasticity arises because of random shocks in the relation between stocks and damages. This stochastic relation between damages and the stock means that the regulator never becomes certain about the true value of g. For example, the regulator does not know whether a high level of damages is caused by a large value of g or by a large realization of the random damage shock. The regulator begins with a prior belief on the unknown parameter. As time progresses and he obtains more observations on the pollutant stock and associated environmental damages, he updates his belief on g using Bayesian estimates. With the informative prior on g having a normal distribution, the posterior on g is also normally distributed with the mean given by a weighted average of the prior and the moment estimates. The uncertainty about damages, together with the possibility of learning, greatly complicates the regulator's problem. There are two types of endogenous learning: the learning about abatement costs that are firms' private information, and about the unknown damage parameter. We examine the feedback strategy. The regulator chooses an optimal control in each period. Firms make their decisions and a random shock arrives. The regulator observes firms' emission responses and environmental damages. These observations enable the regulator to update his priors on both firms' abatement cost shocks and the unknown parameter of damages, creating the priors for the next time period. In setting the optimal control, the regulator must consider the effect of his decision on current expected payoff as well as its effect on future state variables (including the future belief on the marginal environmental damage). Because of the uncertainty and learning about marginal damages, the Principle of Certainty Equivalence no longer holds even if both abatement cost function and environmental damage function are linear-quadratic. The regulator's decision rule becomes non-linear in state variables, and depends on the amount of uncertainty. After calibrating the model with climate change studies, we solve the optimal control problem numerically with the embedded stochasticity and endogenous learning. The solution approach approximates the value function by value function iterations with a flexible functional form using neural networks. We use the calibrated model and theoretical results to assess different policies for controlling global warming. Numerical simulations illustrate the sensitivity of the optimal policy choice to changes in key economic parameters, such as the discount rate, the decay rate of the greenhouse gasses, marginal abatement costs and marginal environmental damages. They support previous works suggesting that taxes are likely to be better than quotas in regulating global climate change. However, a higher CO2 concentration in the atmosphere tends to favor the use of quotas. Uncertainties also affect the optimal policy choice. Higher variability in both random environmental damage shocks and abatement cost shocks favor the use of taxes; but more uncertainty about the unknown marginal environmental damage g favors the use of quotas.
