This problem studies the rebound effect, and the possibility of backfire, for home heating. The rebound effect says that an improvement in energy efficiency may save less energy than expected due to a rebound of energy use. For example, if you insulate your home, then is will take less energy to keep the house at the same temperature as before. But you may want to keep it at a higher temperature now, because it is cheaper to do so than it was before. If the rebound effect is strong enough, then the improvement in energy efficiency may actually lead to an increase in energy use, which is known as backfire. Suppose a consumer has a utility function U = - ( T * - T ( e ) ) 2 + c where T * is the ideal temperature, T ( e ) is the actual temperature that depends on the quantity of energy e that is purchased, and c is the consumption of other goods. A simple model for T ( e ) is T ( e ) = T o + e H , where T o is the outside temperature and H is the heat loss of