L. Suppose a consumer's expenditure function is given by the expression m=U-E*-f(p) Where m is the expenditure required to achieve utility level U when pollution emissions to which the consumer is exposed are E, and where f(p) is a function that is increasing in prices of consumption goods p. a) Derive the marginal damage function, write it out, and sketch a diagram that correctly illustrates this marginal damage function. (3) b) Suppose pollution emissions decrease from level E; to Ey. Explain how you would use the expenditure function to measure the decrease in the total damages to the consumer. (2) Write a short paragraph to explain the concept of the Environmental Kuznets curve and provide a very brief overview of the empirical evidence (i.e., conduct a brief literature search and cite at least two academic sources). (3) Derive the conditions for a Pareto Optimum in an economy in which there is only one person with utility function U(y,y) and two industries with production functions x = f(l,) and y = g (ly)where the total amount of land in the economy is L (where L = L, + [,,)). Show your work. Assume all functions have the standard properties. (5) In the same economy as in question 3, assume that the production of y causes an external cost to the x industry. Specifically, assume now that x = f(l,) ay. That is, each unit of y produced causes external damages in the form of & fewer units of x being produced. Show that the condition for a Pareto Optimum becomes MRS = MRT + a where MRS is the marginal rate of substitution between x and y and MRT is the marginal rate of transformation between x and y. Finally, provide an intuitive explanation of this new condition. (5) 5. Suppose two firms, 1 and 2, have marginal abatement cost functions given by MAC, =90 3e;, MAC, =50 e, Where e; is emissions of a pollutant for firm i. a) Suppose the regulator requires each firm to reduce its emissions to one-half of its uncontrolled level. Explain why this is not cost effective and calculate the overall excess abatement cost of this solution compared to the cost-effective solution for achieving the same total level of abatement. (5) b) Now assume the regulator allocates pollution permits to each of the two firms equal to one- half of their uncontrolled levels of pollution (1 unit of pollution requires 1 permit) and allows them to trade permits if they wish. If there is trade, what quantity does each firm buy or sell to or from the other, and at what prig ) e, assuming they trade at the equilibrium price? (5) Suppose two firms, 1 and 2, have marginal abatement cost functions given by MAC; = 1000 2e; MAC, = 1200 2e, And suppose the marginal damage function is MD = 3E where E = e; + e,. a) Find the efficient amount of total emissions and the efficient amounts for each of the two firms. (5) b) Draw a diagram, with clear and accurate labels, to illustrate your solution. (2) Assume the regulator believes the marginal damage function for greenhouse gas emissions per month is relatively constant over the relevant range of emissions. Further assume the regulator doesn't know the exact aggregate marginal abatement cost curve but does know that it lies somewhere between the two extremes shown in the diagram below. Use this diagram to explain whether you would recommend using an emissions tax policy or a cap-and-trade policy to regulate emissions. (5) MAC high MAC low 0 ' 1 Emissions