3. (20 points) Consider a variant of the \"cakeeating\" model from class. Assume there is a representative consumer whose utility is a function of consumption, Cr , such that U(C,) and 6U/6C > 0 . Consumption is only possible in tandem with the use of a depletable energy resource, such that CI : HEr where 6' denotes energy efciency. The remaining resource at each period is given as the resource in the previous period minus what was used, so that Rf+1 : R, E! . Finally, assume there exists a renewable resource (a backstop) that becomes available at a price higher than the initial cost of the depletable resource, or more generically, at a time T > t . a. Find the first order conditions for a maximum and provide an economic interpretation. What can you say about the price of depletable resource over time? Name: b. Now, assume pollution occurs when the depletable resource is consumed, such that F; : JOE: +(1 6 ) PH , where I? is the amount of the pollutant, p is the emission coefficient associated with consumption of the depletable resource, and 6 is the decay rate of the pollutant. If the consumer intemalizes the negative pollution extemality, so that U (C! , R) with 6U/6P