Economics Dynamics Problems 70

Economics Dynamics Problems 70 - 54 Economic Dynamics...

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54 Economic Dynamics express the diffusion process as dN ( t ) dt = bN ( t )(ln m ln N ( t )) (2.26) or dF ( t ) dt = bF ( t )( ln F ( t )) Suppressing the time variable for convenience, then the two models are ˙ F = ( a + bF )(1 F ) and ˙ F = bF ( ln F ) Pursuing the logistic equation, we can graph ˙ F against F . When F = 0 then ˙ F = a and when ˙ F = 0 then ( a + bF )(1 F ) = 0 with solutions F 1 =− b / a and F 2 = 1 Since ˙ F denotes the rate of diffusion, then the diffusion rate is at a maximum (penetration is at its maximum rate) when ¨ F = 0, i.e., when d 2 F / dt 2 = 0. Differ- entiating and solving for F , which we denote Fp (for maximum penetration rate), we obtain Fp = 1 2 a 2 b implying Np = m · Fp = m ± 1 2 a 2 b ² = m 2 am 2 b In order to Fnd the time tp when F ( tp ) is at a maximum penetration rate, we must Frst solve for F ( t ). This we indicated above. Since we need to Fnd the value of t satisfying F ( t ) = Fp , then we need to solve 1 e ( a + b ) t 1 + ( b / a ) e ( a + b ) t = 1 2 a 2 b for t , which we can do using a software package. This gives the time for the maximum penetration of
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