ECON302_Assignment3_F11_Sol_Full

ECON302_Assignment3_F11_Sol_Full - ECON 302 Macroeconomic...

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Page 1 of 18 Pages ECON 302 Macroeconomic Theory 2 Instructor: Sharif F. Khan Department of Economics University of Waterloo Fall 2011 Suggested Solutions to Assignment 3 (Optional) Total Marks: 80 Marks Read each part of the question very carefully. Show all the steps of your calculations to get full marks, unless it is mentioned otherwise. 1. [45 Marks] Consider a closed economy with a representative consumer, a representative firm and a government. The representative consumer is endowed with h hours of time. Assume that the consumer has the preferences over the consumption good, c , and leisure, l , described by the following utility function. ( ) . ln ln , l C l C U + = She faces a market real wage rate, w , per hour. The consumer is assumed to have h hours of time available, which can be allocated between leisure time, l , and time spent working (or labour supply), denoted by s N . She receives π units of the consumption good as dividend income from the representative firm. Suppose that the government imposes a proportional income tax on the representative consumer. That is, for each w she earns on the market, she pays tw units of consumption good to the government where t is the income tax rate, and 0 < t < 1. There are no lump- sum taxes, or . 0 = T The representative firm is a perfectly competitive firm. Assume that the representative firm produces the consumption good only with labour as an input. It has the following technology to produce the consumption good: , d zN Y = where Y is the aggregate output of consumption goods produced by the firm, z is the exogenous total factor productivity and d N is the hours of labour employed by the firm. The firm faces a market real wage rate, w , per hour.
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Page 2 of 18 Pages The government purchases exogenously fixed G units of consumption goods from the firm. It fully finances this purchase with proportional income taxes collected from the consumer. That is, the government purchases are equal to taxes, in real terms. a) Define a competitive equilibrium for this economy. [5 marks] In this economy a competitive equilibrium is a set of endogenous quantities, C (consumption), N s (labour supply), N d (labour demand), t (income tax rate), Y (aggregate output), and an endogenous real wage, w , such that the exogenous variables G (government spending), z (total factor productivity), and h (the total hours of time available), the following conditions are satisfied: a. The representative consumer chooses C (consumption) and N s (labour supply) to make himself or herself as well off as possible subject to his or her budget constraint, given w (the real wage), t (income tax rate) and π (dividend income). b. The representative firm chooses N d (labour demand) to maximize profits ( d wN Y - = π ) subject to d zN Y = , given z (total factor productivity) and w (the real wage). In equilibrium, the profits that the representative firm earns must be equal to the dividend income that is received by the consumer.
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