World income grows fast without verifiable climate-change impacts on the economy. The growth spell can end if climate impacts turn real but this can take decades to learn. We develop a tractable stochastic climate-economy model with a hidden-state impact process to evaluate the contributions of the expanding economy and changing impact beliefs to the social cost of carbon. Taking a dataset of estimates for the social cost as a representation of beliefs, we assess how robust climate policies are to the delays of hard information. The carbon price should rise with income to the next century, even without observed impacts. The carbon price should grow faster than the economy as long as climate warming is not enough for generating impacts that are informative about the true social cost.