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Economics Dynamics Problems 121

# Economics Dynamics Problems 121 - t P t is given by P t =...

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Discrete dynamic systems 105 then aS = n 1 i = 0 a n i S aS = (1 a ) S = 1 a n S = 1 a n 1 a and y n = a n y 0 + b ± 1 a n 1 a ² Combining these two we can summarise case B as follows y n = y 0 + b na = 1 a n y 0 + b ± 1 a n 1 a ² a ±= 1 (3.18) These particular formulas are useful in dealing with recursive equations in the area of Fnance. We take these up in the exercises. These special cases can be derived immediately using either Mathematica or Maple with the following input instructions 5 : Mathematica RSolve[{y[n+1]==y[n]+b, y[0]==y0},y[n],n] RSolve[{y[n+1]==a y[n]+b, y[0]==y0},y[n],n] Maple rsolve({y(n+1)=y(n)+b, y(0)=y0},y(n)); rsolve({y(n+1)=a*y(n)+b, y(0)=y0},y(n)); 3.6 Compound interest If an amount A is compounded annually at a market interest rate of r for a given number of years, t , then the payment received at time
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Unformatted text preview: t , P t , is given by P t = A (1 + r ) t On the other hand, if it is compounded m times each year, then the payment received is P t = A ³ 1 + r m ´ mt If compounding is done more frequently over the year, then the amount received is larger. The actual interest rate being paid, once allowance is made for the com-pounding,iscalledthe effectiveinterestrate ,whichwedenote re .Therelationship between re and ( r , m ) is developed as follows A (1 + re ) = A ³ 1 + r m ´ m i . e . re = ³ 1 + r m ´ m − 1 It follows that re ≥ r . 5 See section 3.13 on solving recursive equations with Mathematica and Maple ....
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