Most models of optimal climate change policy are complex and opaque. In this paper, it is argued that the convexity of climate damage and mitigation cost function provides a basis for the derivation of simple and robust estimates of optimal stabilization targets and carbon prices. For all but a few extreme assumptions, the optimal carbon price is between $40 and $75. Similarly, for a wide range of parameter values the optimal target is between 425 ppm and 475 ppm. In all simulations, the total cost of mitigation is below 5 per cent of income, and in most cases substantially below.