4. 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 +, where T is the outside temperature and H is the heat loss of the home. Let the price of energy be given by p and the price of other goods be normalized to one. The consumer has income I. a) Formulate the consumer's utility maximization problem. The choice variables are e and c. b) Take the first-order conditions. Solve the first order conditions for e as a function of T", T°, H, and p. c) Suppose T = 72, T° = 32, H = 60, and p = 1. What is the value of e selected by the consumer? What is the temperature of the home? d) Now suppose that we insulate the home, so that H decreases to H = 50. If the consumer leaves the temperature at the same value as in part e, how much energy 2 does the consumer use? If, however, the consumer resolves their optimization problem given the new value of H, what is the new temperature and how much energy do they use? Does backfire occur in this case? e) Sketch a graph of e as a function of H for T = 72, T = 32, and p = 1. Show the values for H such that we get backfire for a small decrease in H.