Transcript Ben-Shahar & Sulganik

‫הכנס השנתי של האיגוד הבינלאומי למדע האזור – הסניף הישראלי‬
April 15, 2008
How Do We Determine Which Housing
Market Allows Greater Mobility?
Danny Ben-Shahar
Technion – Israel Institute of Technology
and
Eyal Sulganik
The Interdisciplinary Center, Israel
MOTIVATION
How do we determine which housing
market allows greater mobility?
Rank the mobility in a given market over time (time
series)
Rank the mobility among markets at a given time-period
(cross section)
INTUITION
Consider an information matrix P where the current
states are considered as signals about the past states
P=
An entry Pij of an information matrix P is the conditional
probability that state sj has emerged as a signal from a
past state si (where i refers to a row and j refers to a
column in the matrix).
INTUITION
According to Blackwell (1953), information matrix P is
considered “more informative than” Q if there exists a
stochastic matrix R such that Q=PR. For example,
Q=PR
P=
Q=
R=
Intuition: R adds noise to P, i.e., Q is a randomized version
of P and, hence, P is more informative than Q.
INTUITION
Suppose, instead, that P is a transition (mobility)
matrix:
P=
An entry Pij of a mobility matrix P is the conditional
probability that a vacancy starting at housing status si
will end up at housing status sj at the end of the period.
Then,…
INTUITION
Mobility matrix Q is considered “more mobile than” P if
there exist a stochastic matrix R such that Q=PR. For
example,
Q=PR
P=
Q=
R=
Intuition: R adds noise to P and therefore the mobility in Q is less
predictable. In other words, the future state in Q is more independent of
current states and, hence, Q is associated with a greater opportunity for
mobility. [See Ben-Shahar and Sulganik (Economica, forthcoming)]
OBJECTIVE
Explore the implications of the proposed mobility
measure to “vacancy chains;”
Explore possible links between the proposed mobility
measure and other mobility measures that appear in the
literature.
OUTLINE OF PRESENTATION
Brief literature review;
Background – vacancy chains;
Selected results;
Summary.
LITERATURE REVIEW
On mobility measures:
Prais (1955), Shorrocks (1978), Brumelle and Gerchak
(1982), Conlisk (1990), Dardanoni (1993), Parker and
Rougier (2001), and Ben-Shahar and Sulganik
(forthcoming).
LITERATURE REVIEW
On vacancy chains in the general literature:
Robson et al. (1999) and Lanaspa et al. (2003) in urban
studies;
Felsenstein and Persky (forthcoming) in labor studies;
Weissburg et al. (1991) in ecology;
Chase and Dewitt (1988) in life science;
Sorensen (1983) in education systems;
Stewman (1988) in criminology;
Chase (1991) presents an overview of vacancy chain
literature.
LITERATURE REVIEW
On vacancy chains in real estate:
Kristof (1965), Adams (1973), and Watson (1974) are
among the firsts to consider vacancy chains emerging
from new construction;
Lansing et al. (1969), racster et al. (1971), and
Brueggeman et al. (1972) were among the first to
suggest the use of vacancy chain models in order to
assess the effectiveness of possible housing policy
programs;
Others: Marullo (1985); Hua (1989); and Emmi and
Magnusson (1995).
BACKGROUND
Vacancy chains:
Given the vacancy transition matrix P,
P=
Suppose s3 is the only absorbing state, then
=
and…
BACKGROUND
is the vacancy chain.
An entry
of the matrix
represents the expected
number of times that a vacancy emerging from state i
will appear in state j before it is absorbed.
RESULTS
Proposition 1: For any two triangular vacancy
transition matrices Q and P, if Q is more mobile than P
(i.e., Q=PR), then
for all i.
RESULTS
Example:
Suppose that
P=
Q=
R=
such that Q=PR
and thus Q is more
mobile then P.
RESULTS
P=
=
RESULTS
Q=
=
RESULTS
RESULTS
Proposition 2: For doubly stochastic transition
matrices P and Q, if Q is more mobile than P (i.e.,
Q=PR) and R is doubly stochastic, then the sum of the
entries in the column of vacancy chain of RP is greater
or equal to the sum of the entries in the respective
column of the vacancy chain of Q.
RESULTS
Proposition 3: If
is a strictly row diagonally
dominant matrix (i.e.,
vacancy chain matrix
), then the
is a strictly diagonally
dominant of its column entries (i.e.,
and j).
for all i
RESULTS
Corollary: If
for all i (i.e., any vacancy that
emerges in status i is always associated with a change
of status), then
columns (that is,
is strictly diagonally dominant in its
for all i and j).
RESULTS
Proposition 4: For any two triangular transition
matrices Q and P, if Q is more mobile than P (i.e.,
Q=PR), then
.
Following Conlisk and Sommers (1979), Shorrocks
(1978), and McFarland (1981), a greater second
largest eigenvalue
is associated with a greater
speed of convergence of a transition matrix to its
equilibrium (i.e., to a constant row matrix).
RESULTS
Proposition 5: For any normal transition matrices P and
Q and a doubly stochastic matrix R, if Q is more mobile
than P (i.e., Q=PR), then
.
Consistent with Parker and Rougier (2001) mobility measure.
Summary
We develop a link between the literature on mobility
measures and the vacancy chain literature;
We derive implications of the mobility measure Q=PR for
vacancy chains;
THE END