Consider a variant of the “cake-eating” model from class. Assume there is a representative consumer whose utility is a function of consumption, C t , such that U ( C t ) and U c > 0. Consumption is only possible in tandem with the use of a depletable energy resource, such that E t = θC t where θ reflects energy efficiency. The remaining resource at each period is given as the resource in the previous period minus what was used, so that R t+1 = R t – E t . 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 the depletable resource over time? b. Now, assume pollution occurs when the depletable resource is consumed, such that P t = ρE t , where P t is the amount of the pollutant, and ρ is the emission coefficient associated with consumption of the depletable resource. If we alter consumer utility so that the negative pollution externality is internalized, where U (C t , P t ) with U p < 0 , how will the optimal pricing rule for the depletable resource be different from your answer in part a? What do you think this means for adoption of the renewable resource? c. How does increased energy efficiency impact your answer? Is this at odds with policies that promote renewables? Discuss.
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