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Unformatted text preview: Discrete dynamic systems
where a is a ﬁxed point. Hence
a = f [λa(1 − a)] = λ[λa(1 − a)][1 − λa(1 − a)]
= λ2 a(1 − a)[1 − λa(1 − a)]
It is at this point where Mathematica or Maple is used to solve this equation.
The range for a stable two-cycle is established by solving10
−1 < f (a1 )f (a2 ) < 1
where a1 and a2 are the two relevant solutions. Since
f (x) = λ(1 − x) − λx
then we can compute f (a1 ) f (a2 ), which is a surprisingly simple equation of the
4 + 2λ − λ2
Hence, we have a stable two-cycle if
−1 < 4 + 2λ − λ2 < 1
Discarding negative values for λ, we establish the range to be 3 < λ < 3.449. Given
we have already a1 and a2 solved for any particular value of λ, then we can ﬁnd
these two stable solutions for any λ in the range just established. Thus, for λ = 3.2
it is readily established using Mathematica or Maple, that a1 = 0.799455 and
a2 = 0.513045, which are the same results as those established using a spreadsheet.
For λ < 3 we have a single ﬁxed point which is stable, which again can readily
be established by means of the same spreadsheet. Finally, if λ = 3.84 the system
converges on a three-cycle result with a1 = 0.149407, a2 = 0.488044 and a3 =
0.959447 (see exercise 13).
As an application of the logistic equation, different from its normal application in
population models (see chapter 14), we turn to the issue of productivity growth
discussed by Baumol and Wolff (1991). Let qt denote the rate of growth of productivity outside of the research development industries; yt the activity level of the
information producing industry (the R&D industries); and pt the price of information. The authors now assume three relationships:
(1) Information contributes to productivity growth according to:
(i) yt+1 = a + byt (2) 10 The price of information grows in proportion to productivity in the sector
outside of the R&D industries, so:
pt+1 − pt
= ν qt+1
pt See theorem 3.2, p. 93 and Sandefur (1990, chapter 4). 121 ...
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.
- Fall '11