This question needs a bit more algebra. You do not need to complete it. But if...

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This question needs a bit more algebra. You do not need to complete it. But if you work hard at it you will see the important effect that compounding has. Suppose that Scrooge has some wealth W and some labor income y. Wealth is a stock of money; Scrooge can grow it by investing it at some rate of return r, which yields a flow of capital income rW during the year. Likewise, labor income y is a flow: each year, repeatedly, Scrooge gets a paycheck of y. (a) Suppose that Scrooge's wealth is Wt in year t. Suppose that he saves a positive fraction s of his total income (capital and labor) in that year. Show how you would calculate Scrooge's new wealth W+1 in year t +1. (b) We will start our accounting from some year which we call t = 0. Use part (a) to develop a full equation for Scrooge's wealth, starting all the way from Wo at date 0, showing that W,+1 = [Wo+ ] (1 + sr)*+1 (1) r (c) Now suppose that Scrooge and Stooge both start with the same wealth, have the same income y year after year, and the same rate of savings s. If Scrooge earns a rate of return pl on his wealth and Stooge earns r2, what will happen to the ratio of Scrooge's wealth to Stooge's wealth as the years go by? Use Equation (1) even if you could do not derive it-just assume it for this question. (d) Again, suppose that Scrooge and Stooge both start with the same wealth and this time they have the same savings rate as well as well as the same rate of return on their wealths. But Scrooge has twice the labor income of Stooge. Now what will happen to the ratio of Scrooge's wealth to Stooge's wealth as the years go by? Once again, use Equation (1). (e) Why do vou think that the answers to (b) and (c) differ so significantlv? This question needs a bit more algebra. You do not need to complete it. But if you work hard at it you will see the important effect that compounding has. Suppose that Scrooge has some wealth W and some labor income y. Wealth is a stock of money; Scrooge can grow it by investing it at some rate of return r, which yields a flow of capital income rW during the year. Likewise, labor income y is a flow: each year, repeatedly, Scrooge gets a paycheck of y. (a) Suppose that Scrooge's wealth is Wt in year t. Suppose that he saves a positive fraction s of his total income (capital and labor) in that year. Show how you would calculate Scrooge's new wealth W+1 in year t +1. (b) We will start our accounting from some year which we call t = 0. Use part (a) to develop a full equation for Scrooge's wealth, starting all the way from Wo at date 0, showing that W,+1 = [Wo+ ] (1 + sr)*+1 (1) r (c) Now suppose that Scrooge and Stooge both start with the same wealth, have the same income y year after year, and the same rate of savings s. If Scrooge earns a rate of return pl on his wealth and Stooge earns r2, what will happen to the ratio of Scrooge's wealth to Stooge's wealth as the years go by? Use Equation (1) even if you could do not derive it-just assume it for this question. (d) Again, suppose that Scrooge and Stooge both start with the same wealth and this time they have the same savings rate as well as well as the same rate of return on their wealths. But Scrooge has twice the labor income of Stooge. Now what will happen to the ratio of Scrooge's wealth to Stooge's wealth as the years go by? Once again, use Equation (1). (e) Why do vou think that the answers to (b) and (c) differ so significantlv?

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a We can calculate Sc roo ge s new wealth in the following way W 1 W y 1 r W s c where c is Sc roo ge s total consumption in year t Sc roo ge s new wealth in year t 1 is equal to his old wealth plus h... View the full answer