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# Econ 101 Javier Birchenall Due date: Tuesday, October 15, 2013 Problem set 2: Labor supply and European vacation The following problem set will...

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Econ 101 Javier Birchenall Due date: Tuesday, October 15, 2013 Problem set 2: Labor supply and European vacation The following problem set will acquaint you with some basic analysis for the labor market, and second, it will make you familiar with an important discussion in macroeconomics, the role of government taxes in labor supply. Plan to spend 8 hours (maybe more) on these questions. Please return the problem set on time (next Tuesday during class), remember you have the chance to miss one and only one, use it wisely. If you have any problems regarding the questions please contact me immediately. Good luck! Please remember to show your work and write your section. I. The labor supply I (35%) This problem considers the decisions of a consumer whose preferences are given by: (   )= + , in which is the quantity of consumption and is the quantity of leisure. The consumer faces two constraints. The time constraint is given by + =1 with as the time spent working (or the labor supply). Notice that we have assumed =1 without loss of generality. We have also assumed that  1 .T h em a i n advantage of working is the wages consumers receive. (a) What is the marginal rate of substitution between consumption and leisure,   ?D r a w t h e indi f erence curves for this utility function. Consumers take wages as given (outside of their control) and obtain wage income equal to  .T h e budget constraint is: =  + , with as divided income and as the real quantity of taxes paid to the government. The problem of the consumer is to decide how many hours to work subject to feasibility. To describe the problem, f rst replace the time constraint into the budget constraint: = (1 )+ , and second, substitute the budget constraint into the objective function. In other words, the problem of the consumer becomes f nding in order to maximize: (   )= { (1 )+ } + ,( * ) with the f rst term simply being . (The Appendix in the book has an alternative method for solving these optimization problems but we will follow a simple substitution as long as it is feasible.) (b) Recall that the consumer decides to equate the   to the wage .U s ing   = , show that the number of hours of work is given by: ( )=1 1 ( 1) , and that ( ) is increasing in the wage. Moreover, show (graphically or formally) that the income e f ects on leisure or labor supply are zero. Remember, that leisure and labor supply are related through thet imeconstra int , + =1 .
II. The labor supply II (35%) Assume now that the utility function of the consumer is di f erent and given by: (   )= + , which di f ers from our previous case because now (   ) is strictly concave in . Because there are no changes in the budget constraint, the set up of the problem is almost identical to the previous case. For instance, once we substitute the budget constraint into the objective function we obtain (   )= ( (1 )+ ) + , (**) (a) Show that if   = , the solution is ( (1 )+ ) 1 = 1 . (b)Assumefortheresto fth isprob lemthat =0 5 and show that leisure in this case is: = + 2 + . and f nd the labor supply. Are income e f ects positive? III. Distortionary taxation and the European vacation (30%) The budget constraint we considered thus far assumes only lump-sum taxes. In reality, lump-sum taxation is used very rarely. One of the reasons is that people are not identical and so the requirement that all people pay the same amount is not feasible. (Some people earn nothing and so are not able to pay any positive amount.) An alternative policy is a proportional labor income tax, where taxes are equal to a f xed fraction , 0 1 of the income. The budget constraint under these taxes is: = (1 )( )+ . (a) Using the notions of income and substitution e f ects, describe the impact of changes in a proportional labor income tax on individual’s behavior. (You may want to use the discussion in pages 114-117 from the textbook as a guide.) Using the information from the table below and a simple macroeconomic model, Prescott (2004) has argued that Europeans supply less labor because there is a much larger wedge in most European countries between what a worker is paid and what that worker actually gets to keep after taxes are taken out. This tax wedge, argues Prescott (2004), distorts the trade-o f people make between consumption and leisure by making consumption more expensive. And since people work, ultimately, to earn money to pay for consumption goods, they will supply less labor if consumption goods become relatively more expensive. The cheaper alternative: leisure. Tax rate (%) Hour of work per week per person Germany 59 19.3 France 59 17.5 Italy 64 16.5 Canada 52 22.9 UK 44 22.8 Japan 37 27.0 US 40 25.9 Source: Prescott (2004), “Why Do Americans Work So Much More Than Europeans?” (b) Using the f ndings from question (a) discuss the role of income and substitution e f ects in Prescott (2004) analysis. You may f nd the discussion in pages 117-118 of the textbook useful. 2

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