Part I of this paper compares and contrasts a simple carbon tax, a rebated carbon tax, and an output-compensated carbon tax, using the electric power industry to illustrate the differences. While these “price instruments” are serious policy proposals, the goal of this paper is to explore the underlying microeconomic theory and, in particular, to highlight some important properties of the compensated demand curves that describe their effects. Part II of the paper explains the mathematical duality between those three price instruments and their corresponding quantity instruments: auctioned cap-and-trade and allocated cap-and-trade, both of which impose a constraint on the quantity of emissions, and emissions-rate-cap-and-trade (also called offset trading), which imposes a constraint on the emission-intensiveness of industrial output. Keeping in mind the properties of compensated demand curves, Part III of the paper argues that the mathematical form of their regulatory constraints can help explain many of the typical differences between economic and social regulation – including the tendency of economic regulatory agencies to be multi-headed “independent” agencies, their inclination to use more adjudication and formal rulemaking than is typical of social regulators, and their troubling susceptibility to agency capture. A mathematical appendix explains the six flavors of compensated demand curve and the shadow prices that shape them.