Economics 370 Microeconomic Theory Problem Set 3 Answer Key 1) Barbara’s only source of income is from working. She can work as many hours per day as she wishes (up to a maximum of 24 hours) at a fixed wage rate of $10 per hour. a) Initially assume that there is no income tax. Draw Barbara’s budget constraint. (Below, in blue) b) Now suppose that the government introduces a tax at a rate of 50 cents in the dollar. Suppose that leisure is a normal good and that the tax ends up reducing Barbara’s hours worked. Show in your diagram the effect of the tax on Barbara’s budget constraint and possible indifference curves for her initial and final consumption. Her new budget constraint is in solid green. 24 240 120 c) Sketch Barbara’s Marshallian labor supply curve and relate your answer carefully to your budget constraint diagram. Also show Barbara’s Hicksian labor supply curve and relate your answer to your budget constraint diagram. In this range, Marshallian labor supply (shown below in blue) is clearly increasing as wages increase. The Hicksian labor supply curve, (the amount labor increases with wages holding utility constant) shown below in green, is always increasing. It is clear from the diagram that the Hicksian labor supply curve is steeper than the Marshallian labor supply curve.Marshallian Hicksian Labor wage d) Now suppose that the government decides to make the first $40 per day (or 4 hours worked) tax free. Beyond this point income is taxed at a rate of 50 cents in the dollar. On a separate diagram, draw Barbara’s budget constraint. Assume that Barbara chooses to work 8 hours per day. How much tax would the government collect? Below, in blue e) Now suppose that the government replaces the tax analyzed in part (d) with a 25 per cent tax rate on all income. Which tax would Barbara prefer? Which would raise more tax revenue? This is our now usual story. If Barbara can afford her earlier bundle (and she can) she cannot be worse off. If indifference curves are smooth (as I drew them below) than the new budget line cuts through her previous indifference curve, and the new budget line makes her strictly better off. In order to attain her higher utility level, she works more, which makes the government better off too, since it raises more tax revenue. 240 24 20 16 140 2) Ernie lives for two periods and earns fixed amounts of non-interest income of m1 in period 1 and m2 in period 2. He can borrow or lend as much as he wishes at the market interest rate, r0. Assume that for Ernie, consumption in each period isa normal good. Suppose that the interest rate at which Ernie can borrow or lends falls from r0 to r1 where r1 < r0. Say whether the following statements are true or false and explain why. a) If Ernie would have been a borrower at the interest rate r0, he will borrow more at the interest rate r1. True. This is illustrated starting on page 187 of your book. We want to use Slutsky’s equation to demonstrate this. Slutsky’s equation is: ()dmdccmdrdcdrdcs11111−+= . It is important, in this case, to recognize the 1 + r is the price c1 in terms of c2. To replace c1 with c2 in the equation above would give an incorrect answer. Of the terms above, the first, which represents the substitution effect, is always negative. We always assume c1 and c2 are normal goods, so the derivative in the second term is positive. Since Ernie is a borrower, c1 > m1. ()() () () ()+−−−−+=dmdccmdrdcdrdcs11111 So a marginal increase in price reduces borrowing. In our case, price goes down, so borrowing increases. b) If Ernie would have neither borrowed nor lent at the interest rate r0, he will end up borrowing at the interest rate r1. True. Slutsky’s equation gives: ()() () () ()+−−−+=011111dmdccmdrdcdrdcs We have a pure substitution effect. Since price dropped, consumption c1 increases, so Ernie ends up borrowing. c) If Ernie would have been a saver at the interest rate r0, he will lend less at the interest rate r1 and may even end up borrowing. False. Again, Slutsky’s equation gives: ()() () () ()++−−−+=dmdccmdrdcdrdcs11111 The effect is ambiguous, so we cannot say whether saving increases or decreases. 3) Anna earns an exogeneous income of m1 in period 1 and m2 in period 2. The government decides to levy a tax on Anna in period 1 of $X and use this to pay her a pension of $X(1+r) in period 2.a) First consider the case where Anna faces no constraints and can borrow or lend as much as she wishes at the interest rate r. True or False (and explain why): This combined tax and pension scheme would have no effect on Anna’s utility and causes her to save less (or borrow more). True. The budget constraint for Anna before the tax is rmmrcc++≤++112121 The budget constraint after the tax and pension scheme is: ()()()xxrmmrxrxrmmrxrmxmrcc +−++=+++−++=++++−≤++111111121212121, or rmmrcc++≤++112121; which is exactly the budget constraint she faced before. Since here preferences have not changed and her budget constraint has not changed, her consumption choices will not change. She reacts to the government plan be reducing her savings by exactly the amount the government taxes her income, thus entirely undoing the plan. b) Now consider the case where Anna can lend as much as she wishes at the interest rate r, but is unable to borrow. Also suppose that she would choose to save in the absence of the tax and pension. True or False (and explain why): This combined tax and pension would have no effect on Anna’s utility and causes her to save less. False. Or, more correctly, maybe. Here it is best to draw a graph. If her preferred consumption involves 0 < c1 ≤ m1 – x (that is, if her preferred consumption is to the left of the pension endowment), then nothing will change. However, if her preferred consumption is between the pension endowment and her original endowment, she will not be able to completely undo the pension scheme. Consumption will be at the pension endowment. m1 Pension endowmentoriginal endowmentm1 – xc1 c2 m2 + (1+r)x m24) Bert has wealth of $50,000 and owes a moneylender $10,000. The terms on which he has borrowed specify that if he does not repay the loan by today he must pay $40,000. The moneylender, however, is absent minded and Bert believes there is an eighty percent probability that the moneylender has forgotten about the loan. a) True or False (and explain why): If Bert repays the loan, then he cannot be risk