final10 - Final Exam EC720.01 Math for Economists Boston...

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Final Exam EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2010 Due Tuesday, December 14, 2010 at 11:00am This exam has three questions on five pages; before you begin, please check to make sure your copy has all three questions and all five pages. Each of the three questions is worth 20 points; as indicated at the beginning of the semester, this 60-point final will account for 60 percent of your course grade and the problem sets will account for the remaining 40 percent of your course grade. This is an open-book exam, meaning that it is fine for you to consult your notes, textbooks, and other references when working on your answers to the questions. I expect you to work independently on the exam, however, so that the answers you submit are yours and yours alone. 1. Growth and Pollution, Part I This question asks you to use the Kuhn-Tucker theorem to solve a static constrained opti- mization problem that can be used to think about patterns of pollution and environmental quality seen when looking across a sample of countries at di ff erent stages of economic devel- opment. Suppose, in particular, that a country has k > 0 units of physical capital that can be used to produce output of a single consumption good using any one from a continuum of technologies indexed by z [0 , 1]. Lower values of z represent “cleaner” technologies that generate less pollution and higher values of z represent “dirtier” technologies that generate more pollution. People dislike pollution, but they also like consumption; and because dirtier technologies can be used to produce output at a lower cost, a trade-o ff arises between economic growth and environmental quality. To make these ideas more specific, suppose that the country’s k units of capital, when used with technology z , produce c = Akz units of output, where A > 0 is a parameter governing the overall level of productivity across all available technologies. Hence, according to this specification, more capital used with dirtier technologies yields more output. Suppose also, however, that the same k units of capital, when used with the same technology z , generate p = Akz β units of pollution, where β > 1 is a parameter governing how the use of dirtier technologies causes environmental quality to deteriorate. Suppose, finally, that a representative consumer in this country gets utility u ( c ) = c 1 σ 1 1 σ 1
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from consuming c units of output but su ff ers disutility v ( p ) = B γ p γ when the level of pollution is given by p , where the preference parameters satisfy σ > 0, B > 0, and γ > 1. A social planner for this economy takes the stock of physical capital k as given, and chooses the levels of consumption c and pollution p to maximize the representative consumer’s total utility subject to the constraints imposed by the description of the technologies from above.
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