velp hr

Exercises 151 4. The government wishes to raise an amount 3/4 in tax revenue from this household. Which tax rate(s) can the government use. Assume that the government is benevolent (that is, nice). Which tax rate does the government use? Exercise 13.5 (Hard) Tammy lives for two periods. In the first period of life she works / hours at a wage of w per hour. In the second period of life she is retired and supplies no labor. There is a perfect bond market on which Tammy saves an amount b in the first period of life, which earns a net interest rate of r = 0 (for a gross interest rate 1 + r = 1), so Tammy's savings pay off (1 + r)b = b in the second period of life. Tammy has preferences over consumption sequences {c1, c2) and effort ( as follows: "((a,),9) = va+ ve-e. The government taxes Tammy at a rate n on income earned in the first period of life, we. Tammy owes no tax in the second period of life. 1. Write down Tammy's budget constraint in each period of her life, substitute out the savings term b and show that Tammy's present-value budget constraint is: atoza+es (1-nje. 2. Calculate Tammy's optimal choices of consumption, work, and savings: c, ez, f, and b, as a function of the tax rate n- 3. How much revenue does the government raise as a function of the tax rate n, T(). Find r, the tax rate at which government revenue is maximized. What is the maxi- mum amount of revenue the government can raise (that is, what is 71()))? 4. Is there a Laffer curve? Exercise 13.6 (Hard) Use the same preferences and technology as in Exercise (13.5) above, except that here we call the government tax rate 2 instead of n. The government will allow Tammy to exempt her savings b from taxes (as in a 401(k) plan), so she owes tax at a rate ry in the first period only on the portion of her income that she does not save, we - b. Tammy still owes no tax in the second period of life. If Tammy saves an amount & she consumes (1 - 12)(wf - b) in period 1 and b in period 2. 1. Write down Tammy's present-value budget constraint. How is it different from the one you calculated in part (1) of Exercise (13.5) above? 2. Solve Tammy's optimization problem. What are her optimal choices of consumption, work, and savings c1, c2, f, and b as a function of the tax rate ry? 152 The Effect of Taxation 3. Now how much revenue does the government raise as a function of the tax rate r2 (call the revenue function 72(12)? What is the maximum amount of revenue the gov- ernment can raise? 4. Is there a Laffer curve? 5. How do your answers differ from those in Exercise (13.5)? Why?Exercises Exercise 14.1 (Easy) Supply the following facts. Most can be found in the Barro textbook. 1. What was the ratio of nominal outstanding public debt to GNP for the U.S. in 1996? 2. About what was the highest debt/GNP ratio experienced by the U.S. since 1900? In what year? 3. According to Barro, about what was the highest marginal tax rate paid by the "aver- age" American since 1900? In what year? 4. Illinois has a standard deduction of about $2000. Every dollar of income after that is taxed at a constant rate (a "flat" tax). What is that rate? Exercise 14.2 (Moderate) The government must raise a sum of G from the representative household using only excise taxes on the two goods in the economy. The household has preferences over the two goods of: U(2, 72) = In(21) + 72. Exercises 171 Variable Definition B Real end-of-period savings by the household sec- tor. Real par value of outstanding end-of-period gov- ernment debt. B2 Initial stock of government debt, B2, = 0. Ct, ct Aggregate (household) consumption att. Real government spending at t. Aggregate (household) income at t. Real net return on end-of-period debt. C Present discounted value of consumption stream, {Geo- G Present discounted value of government spend- ing stream, {Gift-0. Present discounted value of tax revenue stream, W Annuity value of income stream, (Yi}co- V Present discounted value of income stream, H(a; ) Government tax policy mapping household ac- tion a and a vector of tax parameters . into a tax bill (see Chapter 13). T () Government revenue as a function of tax pol- icy parameters / when the household chooses its best response, (max (see Chapter 13). Utility from a consumption stream. u(C) One-period utility from consumption of C in pe- riod t. V({n)) Household's indirect utility given a stream of ex- cise taxes {n), and an income stream (Yiji, with annuity value W. Table 14.1: Notation for Chapter 14. Note that, with the assumption that Y', = N, variables denoted as per-capita are also expressed as fractions of GDP. The household has total wealth of M to divide between expenditures on the two goods. The two goods have prices p and p2, and the government levies excise taxes of t and to. The government purchases are thrown into the sea, and do not affect the household's utility or decisions. Determine the government revenue function 7(t1, tz; pi, p2, M). Determine the household's indirect utility function in the presence of excise taxes, V(pi + 1, p2 +t2, M).Exercise 14.3 (Easy) Assume that a household lives for two periods and has endowments in each period of ( = 1, 12 = 1+ r). The household can save from period t = 1 to period t = 2 on a bond market at the constant net interest rate r. If the household wishes to borrow to finance consumption in period t = 1 and repay in period t = 2, it must pay a higher net interest rate of r' > r. The household has preferences over consumption in period t, cy, of: V((a, )) = u(c) + u(c). Assume that u' >0 and " 0 and 72 > 0. Draw a set of axes. Put consumption in period t = 2, c2, on the vertical axis and consumption in period t = 1 on the horizontal axis. Draw the household's budget set. 3. Now assume that 7 1 = 0 and 72 = (1 + r) G. Draw another set of axes and repeat the previous exercise. 4. Now assume that 71 = G and 72 = 0. Draw another set of axes and repeat the previous exercise. 5. What tax sequence would a benevolent government choose? Why? Exercise 14.4 (Easy) Assume that the government can only finance deficits with debt (it cannot print money, as in the chapter). Assume further that there is a constant real interest rate on government debt of r, and that the government faces a known sequence of real expenditures (G, )- The government chooses a sequence of taxes that produces revenues of (7:}, such that: To = Go -1, and: T: = Gt, for all t = 1, 2, ... , DO. Find the path of government debt implied by this fiscal policy. Does it satisfy the transver- sality condition? Why or why not?Exercise 14.3 (Easy) Assume that a household lives for two periods and has endowments in each period of ( = 1, 12 = 1+ r). The household can save from period t = 1 to period t = 2 on a bond market at the constant net interest rate r. If the household wishes to borrow to finance consumption in period t = 1 and repay in period t = 2, it must pay a higher net interest rate of r' > r. The household has preferences over consumption in period t, cy, of: V((a, )) = u(c) + u(c). Assume that u' >0 and " 0 and 72 > 0. Draw a set of axes. Put consumption in period t = 2, c2, on the vertical axis and consumption in period t = 1 on the horizontal axis. Draw the household's budget set. 3. Now assume that 7 1 = 0 and 72 = (1 + r) G. Draw another set of axes and repeat the previous exercise. 4. Now assume that 71 = G and 72 = 0. Draw another set of axes and repeat the previous exercise. 5. What tax sequence would a benevolent government choose? Why? Exercise 14.4 (Easy) Assume that the government can only finance deficits with debt (it cannot print money, as in the chapter). Assume further that there is a constant real interest rate on government debt of r, and that the government faces a known sequence of real expenditures (G, )- The government chooses a sequence of taxes that produces revenues of (7:}, such that: To = Go -1, and: T: = Gt, for all t = 1, 2, ... , DO. Find the path of government debt implied by this fiscal policy. Does it satisfy the transver- sality condition? Why or why not?Exercise 14.3 (Easy) Assume that a household lives for two periods and has endowments in each period of ( = 1, 12 = 1+ r). The household can save from period t = 1 to period t = 2 on a bond market at the constant net interest rate r. If the household wishes to borrow to finance consumption in period t = 1 and repay in period t = 2, it must pay a higher net interest rate of r' > r. The household has preferences over consumption in period t, cy, of: V((a, )) = u(c) + u(c). Assume that u' >0 and " 0 and 72 > 0. Draw a set of axes. Put consumption in period t = 2, c2, on the vertical axis and consumption in period t = 1 on the horizontal axis. Draw the household's budget set. 3. Now assume that 7 1 = 0 and 72 = (1 + r) G. Draw another set of axes and repeat the previous exercise. 4. Now assume that 71 = G and 72 = 0. Draw another set of axes and repeat the previous exercise. 5. What tax sequence would a benevolent government choose? Why? Exercise 14.4 (Easy) Assume that the government can only finance deficits with debt (it cannot print money, as in the chapter). Assume further that there is a constant real interest rate on government debt of r, and that the government faces a known sequence of real expenditures (G, )- The government chooses a sequence of taxes that produces revenues of (7:}, such that: To = Go -1, and: T: = Gt, for all t = 1, 2, ... , DO. Find the path of government debt implied by this fiscal policy. Does it satisfy the transver- sality condition? Why or why not?Exercises Exercise 15.1 (Moderate) Derive the equations for off, of, co", and of * from Section 15.2. See equations 15.3 and 15.4. Exercise 15.2 (Moderate) Pat and Chris work 8 hours each day. They each try to make as much money as possible in this time. Pat can make a jug of wine in 2 hours and a jug of beer in 1 hour. Chris can make a jug of wine in 6 hours and a jug of beer in 2 hours. Pat and Chris are the only producers of wine in this economy. The price of wine is po, and the price of beer is pa. The daily demand for wine is: QW = 11 -2 20 1. Graph the aggregate supply curve for wine. 2. Graph the demand curve for wine (on the same graph). 3. Determine the equilibrium relative price of wine (i.e., the value of pow /pp that causes supply to equal demand). 4. Calculate the equilibrium values of: (i) the amount of wine made by Pat; (ii) the amount of beer made by Pat; (iii) the amount of wine made by Chris; and (iv) the amount of beer made by Chris. 5. Does either Pat or Chris have an absolute advantage in wine production? If so, which does? G. Does either Pat or Chris have a comparative advantage in wine production? If so, which does?Exercises Exercise 17.1 (Easy) In the model of bank runs, explicitly calculate the interest rate on deposits held until period t = 2, ry, when the interest rate on deposits held until period t = 1 is r- Exercises 207 Exercise 17.2 (Moderate) For this problem, assume that there are only two types of potential borrowers: Safe (who comprise a of the population) and Risky (who comprise the remaining 1 -a of the popula- tion). Banks cannot tell the difference between them, and with probability a, a borrower is safe and probability 1 -a a borrower is risky. Safe borrowers have access to safe projects, which pay off as if they succeed and 0 if they fail. Safe projects succeed with probability ps. Risky borrowers have access to risky projects, which pay off an if they succeed and zero if they fail. Risky projects succeed with probability PR. Risky and safe projects have the same expected payoff: PST s = PRTR but the probability of success is lower for risky projects, so pa *s. Both risky and safe projects have public failure, that is, there is no need to audit agents who claim that their project failed. To finance the projects borrowers need a unit of capital from a bank. The bank in turn announces a repayment amount a in the event that the borrower's project does not fail. If the project fails, borrowers owe nothing (they declare bankruptcy). If the project succeeds, borrowers consume their output minus a, if the project fails, borrowers consume zero. Assume that borrowers are risk neutral so that their utility function is just their expected consumption. There is a risk-free interest rate of r that banks must pay to their depositors (thus they have to realize at least 1 + r in expected value on their loan to meet their deposit liability). 1. Write down a bank's balance sheet (in terms of z, r, ps, and pa) assuming that, with probability a the borrower is safe and with probability 1 - a the borrower is risky. 2. Assume that banks compete by offering the lowest value of a that gives them non- negative profits in expectation. Determine the equilibrium interest rate a" (r, a) as a function of the interest rate r and the proportion of safe agents a. 3. Find the expected utility of a safe agent who borrows, Vs(r), as a function of the interest rate r when a is given by a*(r, a). Repeat for a risky agent. 4. Agents stop borrowing if the expected utility of being a borrower falls below zero. Show that if a safe agent decides to borrow, a risky agents will too. Find the critical interest rate r* at which safe agents stop borrowing. At interest rates greater than or equal to this critical value, r 2 r* all safe agents leave the pool, so a = 0. What happens to the equilibrium payment z?Exercise 17.3 (Moderate) Consider the model of costly audits again. Now suppose that intermediaries gain access to a technology which allows them to extract more from each borrower (that is, for each value of announced repayment a and audit cost y, suppose .(2, 7) shifts up). What happens to 208 Financial Intermediation the demand schedule of capital? What happens to the supply schedule of capital? What happens to the equilibrium interest rate? What happens to equilibrium economy-wide output? Are agents made better off or worse off? Exercise 17.4 (Moderate) Yale University costs 1 dollars to attend. After graduation, Yalies (that is, graduates of Yale) either land good jobs paying w or no job at all, paying nothing. The probability of landing the good job is a where a is hidden effort exerted by the Yalie. Yalies are born with wealth a 2 0, and those Yalies born with wealth a 1 to finance the loans. Student borrowers who get the good job must repay Yale University some amount z out of their wages w. Student borrowers who do not land the good job pay nothing. All students have preferences over lifetime expected consumption E (c) and private labor effort . of: V(E(c), T) = E(c) - -7 Assume 0 1. Show that her optimal effort a* is a. 2. Now consider poor Yalies, with a