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Chicago State University Economics Mathematics Worksheet

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Problem Set 1 (45 points) ECN 230 Prof. Rowan Shi Winter 2022 I set take-home open-book assignments rather than midterm exams because I think it can be a better chance for students to demonstrate their problem-solving skills — so I invest a lot of trust in your academic integrity. In return, I ask you to have enough trust in your own abilities to not work with anyone else or consult the internet. For this reason: please write the honour code on your solutions. 1. (29 points) Suppose a city’s quality of life is given by u(y, p), where y is the upkeep of the city’s subway network and p is the upkeep of its parks. a) (2 points) How many first order partial derivatives does the function u have? What are they? b) (2 points) Choose one of the first order partial derivatives and explain what it describes, in the context of quality of life and city upkeep. The actual upkeep of the subway network and parks depend on the amount of provincial funding the city receives f as well as how long winter is w. The longer the winter, the harder it is to maintain city amenities. In particular, suppose y = g( f , w) p = h( f , w) c) (6 points) What is the likely effect of provincial funding on city quality of life? Justify your answer mathematically. d) (6 points) What is the likely effect of winter length on city quality of life? Justify your answer mathematically. 1 Now suppose that, if the subway upkeep is y, then the subway system generates total revenue r (y). The premier’s advisors must decide how much funding f to provide the city. They choose f by maximising profits r (y) − f where they know the value of w = w ahead of time and that y = g( f , w). (If it helps, you can imagine w = 78 . However, please still express all answers in terms of w unless the question says otherwise.) e) (2 points) Describe in words what g( f , w) represents (e.g. in the context of provincial funding, winter length, etc). f) (2 points) Write the first order condition of the maximisation problem faced by the premier’s advisors. Finally, suppose that 3 2 g( f , w) = f 4 − w r (y) = y 3 g) (1 point) Write the first order condition of the maximisation problem faced by the premier’s advisors with these functional forms. h) (1 point) This year, winter is ten and a half months long (!), i.e. w = 78 . Show that optimal provincial funding this year is f = 1. i) (5 points) Suppose the advisors know that, next year, winter will be slightly shorter than it was this year. Will optimal funding next year be higher or lower? Justify your answer mathematically, but do not redo the optimisation problem. Hint: you need to compute a derivative here. j) (2 points) Explain the economic intuition of your previous answer. 2. (16 points) Consider an economy with two sectors: manufacturing (m) and services (s). To produce $1 of manufacturing output, the sector requires $a of manufacturing input and $c dollars of services input. To produce $1 of services output, the sector requires $b units of manufacturing input and $d units of of services input. Assume that 0
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