Part 1Question 1Dora lives in an island and her utility function isU =√c +p(24 − N)where c denotes the number of apples she consumes and N denotes the number of hours sheworks in order to collect the apples. She picks up 2 apples per each working hour. Everymorning before starting to pick up the apples Dora gets A > 0 apples “for free” from the sky.(1) Set up the maximization problem of Dora(2) Derive the first order condition(3) Which of the following statements is correct(a) If a=50 than Dora doesn’t work at all.(b) Dora’s working decision does not depend on the size of a.(c) Dora won’t work more than 16 hours.(d) Dora won’t work more than 20 hours.Answer :1) The maximization problem is given by max√c +p(24 − l)subject to c = y = 2l + a2) Substituting the budget constraint into the utility function you can get 2C−(1/2)=(24 − l)−(1/2)and then combining these two equation we can get c = 96 − 4l3) Use c = 96 − 4l = 2l + a to get l = 16 −a6and thus the right answer is (3).Question 2Dora lives in an island and her utility function isU =13ln(c) +23ln(1 − N )where N is the amount of hours put into collecting apples which she consumes (denoted by c).Dora’s production function isc = 2 ∗ N(1) Solve the optimal allocation of hours and consumption (11 points)1Now assume that the government taxes Dora by 20% such thatc = 2 ∗ 0.8 ∗ Ni.e. for every hour worker Dora can now only consume 1.6 apples.(2) Describe in words the income and substitution effect that the tax have(3) Resolve the optimal allocation of hours and consumption in the presence of the tax rate(4) Draw the budget constraint and preferences that corresponds to the “before” and “after”the tax. On the X-axis have hours worked and on the Y-axis have consumptionAnswerThe problem is max13ln(c) +23ln(1 −N)s.t. c = 2N We can write it as max13ln(2N ) +23ln(1 − N) The FOC with respect to N is then131N=2311−N1 − N = 2N N =13and thusconsumption and output equal23.In the case of a tax we rewrite the problem as max13ln(c)+23ln(1−N ) s.t. c = 2∗0.8∗NWe can write it as max13ln(2 ∗ 0.8 ∗ N ) +23ln(1 − N )We get again N =13but this timeconsumption equals c = 2 ∗ 0.8 ∗13=1.63The tax has an income and substitution effect. The income effect is that the worker ispoorer which will increase the amount of hours worked. The substitution effect is that leisure ischeaper (since working is now more expensive) which will reduce the amount of hours worked.In this example both effects cancel each other. As such the indifference curve will intersect atthe same working hours but at vertical lower consumption level.Question 3Mrs. Robinson can use her labor in a production function of bananasY =√NAssume that Mrs. Robinson derives utility from her consumption of bananas and her leisure.Mrs. Robinson used to maximize her utility and work 16 hours a day which allowed her toconsume four bananas per day (i.e. she used to have 8 hours of leisure per day). One day shewas offered to get two bananas for free if she’d adopt a different production function,Y = N(1/4)What can you say about her consumption of bananas and leisure if she accepts the new offer?Will she accept the offer?AnswerMrs. Robinson used to enjoy eight hours of leisure and eat four bananas - this was heroptimal allocation of effort. With the new production she can get the same allocation - so she2cannot be worse of. This implies that this change has no income effect but only a substitutioneffect since the MPL with the new production function is smaller than with the original one.This implies Mrs. Robinson is less efficient (the ?price? of a working hour is higher now sinceshe gets less output for it). Thus the optimal thing to do given the substitution effect is towork less, consume more leisure and eat less bananas. We know she will take the offer becauseshe can always replicate exactly the same allocation of leisure and food she used to have.Question 4Suppose you used to get paid $10 per hour for the first 6 hours you work, and $20 for eachhour of overtime. Given these you decided to work 8 hours. Now your boss decided to changethe wage structure and she will start paying you to pay a fixed wage of $12.5 per hour.1. How will this change the amount of hours you work? Hint: Use the concepts of incomeand substitution effects.(a) You will work more then before(b) You will work less then before(c) You will work the same amount of hours(d) Unclear without more informationAnswerNote that you can keep on working the same amount of hours. So the change has noincome effect. It only has a substitution effect that makes your last working hour lessbeneficial. So you will work less. The answer is B.2. Are you supportive of the change in the compensation policy?(a) Yes(b) No(c) You will be indifferent(d) Unclear without more informationAnswerYes. You can always repeat the same working hours as before. If you choose note to doso then you must be better off. The answer is A.Now suppose your boss decided to change the wage structure but not to $12.5 per hours.Rather instead of paying you $10 per hour for the first 6 hours and $20 for each hour ofovertime she will start paying you to pay a fixed wage of $25 per hour.3. How will this change the amount of hours you work? Hint: Use the concepts of incomeand substitution effects.(a) You will work more then before(b) You will work less then before3(c) You will work the same amount of hours(d) Unclear without more informationAnswerUnclear. At the same allocation of hours you are making more. So the income effectmake you less willing to work. But the substitution effect makes you more willing towork. So the answer is D.4. Are you supportive of the change in the compensation policy?(a) Yes(b) No(c) You will be indifferent(d) Unclear without more informationAnswerYes. You can always repeat the same working hours as before. If you choose note todo so then you must be better off. The answer is A.4Part 2Question 1Assume that a firm produces only with labor according to the following production functionY = A ∗ N where N is the amount of labor hired by the firm and A is the productivity of aworker. The wage rate which the firm is taking as given is W.1. Write down the firm’s maximization problem.2. Which of the following statements is correct for every level of A:(a) The firm will hire for sure some workers(b) The firm might hire no workers but it will not try to hire more than (A/W) amountof workers.(c)