PROBLEM SET 514.02 Principles of MacroeconomicsApril 6, 2005Due April 13, 2005I. Answer each as True, False, or Uncertain, providing some ex-planation for your choice.1. Fluctuations in stock prices may arise from changes in investors’attitudestowards risk.2. A pe rmanent tax cut, all else equal, will have a quantitatively largerexpansionary e¤ect than a temporary one.3. Stock market bubbles may lead to "too much" investment.4. If the return on $1 invested in stocks is greater than the return on $1invested in bonds (and investors are risk neutral), then arbitrage will ensurethat the two returns are equalized through a decrease in the price of stocks.5. The ability to borrow against your future (expected) income is a criticaldeterminant of how closely your current consumption will track your currentincome.II. ConsumptionSuppose you are 22, about to graduate from MIT and take a job thatpromises an annual income of $Y . You expect your salary to increase by g%each year. The annual nominal interest rate is i% and is expected to remainconstant forever. Assume that you are paid once a year, and that you get your…rst paycheck at the end of the …rst year, the second one at the e nd of thesecond year and so on.(i) Suppose you expect to live forever. Assume consumption is constant.When is this constant consumption …nite? Explain what is going on when thecondition (for it being …nite) is not satis…ed.(ii) Assume that you expect to live for n years (i.e., your expected age atdeath will be n+22). When you sign your employment contract, you can chooseone of two options - you can either work for all n years, or you can opt to retireat the age of 60 (n + 22 > 60) in exchange for a single lumpsum payment of $L(at the end of that year). What condition would $L have to satisfy for you tochoose th e voluntary retirement option?(iii) Suppose you are once again faced with the option of choosing retirementat the age of 60, or never at all, and also that consumption is constant for yourlifetime. If you choose to work, you expect to live for n years as in (ii). If youchoose retirement at 60, however, you may expect to live for N n extra years(since working requires physical resources, and leisure partly renews them).Suppose the parameters in (ii) were s uch that you chose voluntary retirement.Is that su¢ cient to ensure that you choose voluntary retirement in this case aswell? If not, why not, and what condition would need to be satis…ed for you tochoose voluntary retirement? (Note : this question is not about "the value oflife", but merely about the distinction between c onsu mption and wealth)1(iv) Now assume that retirement is no longer a matter of choice. You haveto retire at the age of 60, and therefore there is no lump sum payment needed toinduce you to retire. Assume that you expect to live for n years (so you expectto die at the age n + 22, and n + 22 > 60) and once again, that you smoothconsumption over your lifetime.(a) Find an expression for the constant annual consumption.(b) You may assume that this constant consumption is greater thanyour starting salary, so that in the early years you will have to borrow. Denotethe number of years you will nee d to borrow as T . Find an expres sion thatimplicitly de…nes T (Ignore the possibility that T may not be a natural number).Discuss how T depends on n.III. InvestmentA manufacturer is considering buying a machine which costs Vtin real terms,and which promises a pro…ts stream (in real terms) of t+1at the end of the…rst year of operation, t+2at the end of the secon d ye ar, and so on. The futureis known with perfect certainty and a constant real interest rate of r is expectedto prevail for the machine’s life, which you may assume to be in…nite. There isno depreciation.(i) Find the condition under which the investment is worthwhile.(ii) De…ne i as the constant interest rate, which would make the presentdiscounted value of future pro…ts equal to the current cost of buying the machine(this rate is often called the Internal Rate of Return). Find a condition relatingi and r such that the investment is worthwhile. Is this c ondition conceptuallyany di¤erent from the condition you derived in (i)? Why or why
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