Economics Dynamics Problems 56

Economics Dynamics Problems 56 - ln | P d r | = rt c P d r...

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40 Economic Dynamics Looked at from the point of view of a differential equation, we can readily establish that dP dt = rP with solution P ( t ) = P 0 e rt Since P 0 is the initial payment, then P 0 = A in this formulation of the problem. We know that an initial deposit, P 0 , compounded continuously at a rate of r per cent per period will grow to P ( t ) = P 0 e rt Now assume that in addition to the interest received, rP , there is a constant rate of deposit, d . Thus dP dt = rP + d The solution to this differential equation can be found as follows 3 dP dt = r [ P + ( d / r )] dP / dt P + ( d / r ) = r then d dt ln | P + ( d / r ) |= r Integrating both sides ± d dt ln | P + ( d / r ) | dt = ± rdt which leads to
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Unformatted text preview: ln | P + ( d / r ) | = rt + c P + ( d / r ) = ce rt c = e c Therefore P ( t ) = ce rt − ( d / r ) If P (0) = P , then P = c − ( d / r ) and P ( t ) = [ P + ( d / r )] e rt − ( d / r ) = P e rt + ( d / r )( e rt − 1) (2.17) We know that P e rt is the interest paid on the initial deposit of P , so ( d / r )( e rt − 1) is the interest paid on the additional deposit rate, d . 3 An alternative solution method is to use the one outlined in section 2.2....
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