The small, landlocked republic of Bupkestan is divided into three economic classes - Upper, Middle and Lower. Generational economic mobility in Bupkestan is governed by the following (statistical) rules: ⚫ Of the children born into the upper class: 。 75% remain in the upper class. 。 25% move to the middle class. 。0% move to the lower class. ⚫ Of the children born into the middle class: 。 10% move to the upper class. 。 80% remain in the middle class. 。 10% move to the lower class. ⚫ Of the children born into the lower class: (a) 。 0% move to the upper class. 。 65% move to the middle class. 。 35% remain in the lower class. Suppose that Un, Mn and Ln are the percentages of generation n in Bupkestan, in the upper, middle and lower classes, respectively. Find the matrix B, such that Gn+1 = B Gn, where Gn is the vector that describes the distribution of generation n in the three classes: . Un Gn = Mn Ln (b) Show that B is an irreducible (aka regular) stochastic matrix. (c) Suppose that Uo = 0.05, Mo = 0.25 and Lo = 0.7. Find U2, M2 and L2. (d) With Uo, Mo and Lo, as in (c), approximately how will generation n of Bupkestan be distributed in the three classes, when n is sufficiently large? Explain your answer in terms of the steady state (probability) vector of B.
The small, landlocked republic of Bupkestan is divided into three economic classes - Upper, Middle and Lower. Generational economic mobility in Bupkestan is governed by the following (statistical) rules: ⚫ Of the children born into the upper class: 。 75% remain in the upper class. 。 25% move to the middle class. 。0% move to the lower class. ⚫ Of the children born into the middle class: 。 10% move to the upper class. 。 80% remain in the middle class. 。 10% move to the lower class. ⚫ Of the children born into the lower class: (a) 。 0% move to the upper class. 。 65% move to the middle class. 。 35% remain in the lower class. Suppose that Un, Mn and Ln are the percentages of generation n in Bupkestan, in the upper, middle and lower classes, respectively. Find the matrix B, such that Gn+1 = B Gn, where Gn is the vector that describes the distribution of generation n in the three classes: . Un Gn = Mn Ln (b) Show that B is an irreducible (aka regular) stochastic matrix. (c) Suppose that Uo = 0.05, Mo = 0.25 and Lo = 0.7. Find U2, M2 and L2. (d) With Uo, Mo and Lo, as in (c), approximately how will generation n of Bupkestan be distributed in the three classes, when n is sufficiently large? Explain your answer in terms of the steady state (probability) vector of B.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Please solve (a) and (b) show ALL the work including the work of finding the steady state vector if needed
Transcribed Image Text:The small, landlocked republic of Bupkestan is divided into three economic classes -
Upper, Middle and Lower. Generational economic mobility in Bupkestan is governed
by the following (statistical) rules:
⚫ Of the children born into the upper class:
。 75% remain in the upper class.
。 25% move to the middle class.
。0% move to the lower class.
⚫ Of the children born into the middle class:
。 10% move to the upper class.
。 80% remain in the middle class.
。 10% move to the lower class.
⚫ Of the children born into the lower class:
(a)
。 0% move to the upper class.
。 65% move to the middle class.
。 35% remain in the lower class.
Suppose that Un, Mn and Ln are the percentages of generation n in
Bupkestan, in the upper, middle and lower classes, respectively. Find the matrix B,
such that Gn+1 = B Gn, where Gn is the vector that describes the distribution of
generation n in the three classes:
.
Un
Gn =
Mn
Ln
(b)
Show that B is an irreducible (aka regular) stochastic matrix.
(c)
Suppose that Uo = 0.05, Mo
=
0.25 and Lo
=
0.7. Find U2, M2 and L2.
(d)
With Uo, Mo and Lo, as in (c), approximately how will generation n of
Bupkestan be distributed in the three classes, when n is sufficiently large? Explain
your answer in terms of the steady state (probability) vector of B.
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