# ECON 3102 Consider an economy with a corn producer

Department of Economics Intermediate Macroeconomics (ECON 3102)Boerma & EslamiSummer 2016Problem Set 1Department policy requires that all homeworkassignments be typed, except those portions whichare mainly computational/math.1. (20 points) [Williamson (2014), Chapter 2, Problem 6] Consider an economy with acorn producer, some consumers, and a government. In a given year, the corn producer grows 30 million bushels of corn and the market price for corn is $5 per bushel.Of the 30 million bushels produced, 20 million are sold to consumers, 5 million arestored in inventory, and 5 million are sold to the government to feed the army. Thecorn producer pays $60 million in wages to consumers and $20 million in taxes tothe government. Consumers pay $10 million in taxes to the government, receive $10million in interest on the government debt, and receive $5 million in Social Securitypayments from the government. The profits of the corn producer are distributed toconsumers.(a) Calculate GDP using (i) the product approach, (ii) the expenditure approach, and(iii) the income approach.(b) Calculate private disposable income, private sector saving, government saving,national saving, and the government deficit. Is the government budget in deficitor surplus?2. (20 points) [Williamson (2014), Chapter 2, Problem 12] Let Kt denote the quantity ofcapital a country has at the beginning of period t. Also, suppose that capital depreciates at a constant rate d, so that dKt of the capital stock wears out during period t.If investment during period t is denoted by It , and the country does not trade withthe rest of the world, then we can say that the quantity of capital at the beginning ofperiod t + 1 is given byKt+1 = (1 ? d) Kt + It .Suppose at the beginning of year 0 that this country has 80 units of capital. Investment expenditures are 10 units in each of years 0, 1, 2, 3, 4, . . . , 10. The capital stockdepreciates by 10% per year.(a) Calculate the quantity of capital at the beginning of years 0, 1, 2, 3, 4, . . . , 10.(b) Repeat part (a), except assume now that the country begins year 0 with 100 unitsof capital. Explain what happens now, and discuss your results in parts (a) and(b).?3. (30 points) Consider a consumer whose utility function is given by u (C, `) = C +2 `,where C represents the consumption of a good or service, and ` represents the timethat is spent on leisure. Clearly, consumption and leisure cannot be negative. (Whatdoes it mean to consume ?2 apples, after all?) Thus, this multi-variate function’sdomain is [0, +?) × [0, +?).In addition, assume that the consumer faces a budget constraint of the formp · C + (w + ? ) · ` = 4 + w .Here, p represents the price of the consumption good, w represents wage, and ? is atax on (labor income). (You can check that, as ? increases, the cost of resting at home,instead of going to work, rises.)Then, the problem that a consumer – who wants to maximize her utility – faces can beformally states as below:?max u (C, `) = C + 2 · `C,`s.t.p · C + (w + ? ) · ` = 4 + w .(a) Solve this constrained optimization problem when p = 2, w = 4, and ? = 1,using the substitution method. Make sure that you find the maximizer(s) andthe maximum.(b) Find the critical points in the optimization problem using the method of Lagrange multipliers. Check to see whether the critical points constitute maximizers or not.(c) Now, divide the two sides of the budget constraint by p, and redo Part 3a. Is youranswer different from Part 3a? Why?(d) Next, suppose the tax rate increases to 2. What happens to the maximizer(s) andthe maximum?24. (30 points) Consider the problem of a firm that want to maximize its profits by choosing the right amount of labor that it hires, and the right amount of output that itproduces. In particular, the problem that this firm faces is given asmax ? (y, n) = 4 · y ? 2 · ny,ns.t.y = ln (1 + n) .(a) Can you interpret this problem? In particular, what are y and n? What does ?represent? What does the constraint (that firm faces) stand for? What are themarket counterparts of the coefficients in the objective function, 2 and 4?(b) Based on your answer to Part 4a, what is the domain of the objective function?What are the endpoints of this domain?(c) Based on your intuition in Part 4a, what is the role of one inside the natural logfunction (in the constraint of the problem)? In other words, does it make senseto drop it, and write the constraint as y = ln (n), keeping in mind that this is theprofit maximization problem of a firm?(d) Solve the firm problem using the substitution method. Make sure to find themaximizer(s) and the maximum.(e) Find the critical points of the problem, using the method of Lagrange multipliers.Check to see whether the critical points constitute maximizers or not.ReferencesWilliamson, Stephen D. (2014). Macroeconomics. 5th Edition. Prentice Hall.

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