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Economics Dynamics Problems 122

# Economics Dynamics Problems 122 - t = a t − 1 bY t − 1...

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106 Economic Dynamics Figure 3.12. Example 3.6 A bank is offering a savings account paying 7% interest per annum, compounded quarterly. What is the effective interest rate? re = 1 + 0 . 07 4 4 1 = 0 . 072 or 7.2%. If we assume that m is a continuous variable, then given an interest rate of say, 7%, we can graph the relationship between re and m . A higher market interest rate leads to a curve wholly above that of the lower interest rate, as shown in figure 3.12. Returning to the compounding result, if an amount is compounded at an annual interest rate r , then at time t we have the relationship Y t = (1 + r ) Y t 1 . If we generalise this further and assume an additional deposit (or withdrawal) in each period, a t , then the resulting recursive equation is Y t = (1 + r ) Y t 1 + a t 1 Or more generally, we have the recursive equation Y
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Unformatted text preview: t = a t − 1 + bY t − 1 Many problems reduce to this kind of relationship. ±or example, population of a species at time t may be proportional to its size in the previous period, but predation may take place each period. Or, human populations may grow proportionally but immigration and emigration occurs in each period. Solving the recursive equation can be achieved by iteration. Let the initial values be Y and a , respectively, then Y 1 = a + bY Y 2 = a 1 + bY 1 = a 1 + b ( a + bY ) = a 1 + ba + b 2 Y Y 3 = a 2 + bY 2 = a 2 + b ( a 1 + ba + b 2 Y ) = a 2 + ba 1 + b 2 a + b 3 Y Y 4 = a 3 + bY 3 = a 3 + b ( a 2 + ba 1 + b 2 a + b 3 Y ) = a 3 + ba 2 + b 2 a 1 + b 3 a + b 4 Y...
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