How Do We Determine Which Housing How Do We Determine Which Housing

Market Allows Greater Mobility?Market Allows Greater Mobility?

Danny Ben-ShaharDanny Ben-ShaharTechnion – Israel Institute of TechnologyTechnion – Israel Institute of Technology

andand

Eyal SulganikEyal SulganikThe Interdisciplinary Center, IsraelThe Interdisciplinary Center, Israel

הכנס השנתי של האיגוד הבינלאומי למדע האזור – הסניף הישראלי

April 15, 2008

MOTIVATIONMOTIVATION

How do we determine which housing How do we determine which housing market allows greater mobility?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)

INTUITIONINTUITION

Consider an Consider an information matrixinformation matrix PP where 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).

INTUITIONINTUITION

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,

P = Q =

Intuition: R adds noise to P, i.e., Q is a randomized version of P and, hence, P is more informative than Q.

Q=PR

R =

INTUITIONINTUITION

Suppose, instead, that Suppose, instead, that PP is a is a transition (mobility) matrixtransition (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,…

INTUITIONINTUITION

Mobility matrix Q is considered “more mobile than” P if there exist a stochastic matrix R such that Q=PR. For example,

P = Q =

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)]

Q=PR

R =

OBJECTIVEOBJECTIVE

Explore the implications of the proposed mobility Explore the implications of the proposed mobility measure to “vacancy chains;”measure to “vacancy chains;”

Explore possible links between the proposed mobility Explore possible links between the proposed mobility measure and other mobility measures that appear in the measure and other mobility measures that appear in the literature.literature.

OUTLINE OF PRESENTATIONOUTLINE OF PRESENTATION

Brief literature review;Brief literature review;

Background – vacancy chains;Background – vacancy chains;

Selected results;Selected results;

Summary.Summary.

LITERATURE REVIEWLITERATURE REVIEW

On On mobility measuresmobility measures::

Prais (1955), Shorrocks (1978), Brumelle and Gerchak Prais (1955), Shorrocks (1978), Brumelle and Gerchak

(1982), Conlisk (1990), Dardanoni (1993), Parker and (1982), Conlisk (1990), Dardanoni (1993), Parker and

Rougier (2001), and Ben-Shahar and Sulganik Rougier (2001), and Ben-Shahar and Sulganik

(forthcoming).(forthcoming).

LITERATURE REVIEWLITERATURE REVIEW

On On vacancy chains vacancy chains in the general literature:in the general literature:

Robson et al. (1999) and Lanaspa et al. (2003) in urban Robson et al. (1999) and Lanaspa et al. (2003) in urban studies;studies;Felsenstein and Persky (forthcoming) in labor studies;Felsenstein and Persky (forthcoming) in labor studies;Weissburg et al. (1991) in ecology;Weissburg et al. (1991) in ecology;Chase and Dewitt (1988) in life science;Chase and Dewitt (1988) in life science;Sorensen (1983) in education systems;Sorensen (1983) in education systems;Stewman (1988) in criminology;Stewman (1988) in criminology;Chase (1991) presents an overview of vacancy chain Chase (1991) presents an overview of vacancy chain literature.literature.

LITERATURE REVIEWLITERATURE REVIEW

On On vacancy chainsvacancy chains in real estate: in real estate:

Kristof (1965), Adams (1973), and Watson (1974) are Kristof (1965), Adams (1973), and Watson (1974) are among the firsts to consider vacancy chains emerging among the firsts to consider vacancy chains emerging from new construction;from new construction;

Lansing et al. (1969), racster et al. (1971), and Lansing et al. (1969), racster et al. (1971), and Brueggeman et al. (1972) were among the first to Brueggeman et al. (1972) were among the first to suggest the use of vacancy chain models in order to suggest the use of vacancy chain models in order to assess the effectiveness of possible housing policy assess the effectiveness of possible housing policy programs;programs;

Others: Marullo (1985); Hua (1989); and Emmi and Others: Marullo (1985); Hua (1989); and Emmi and Magnusson (1995). Magnusson (1995).

BACKGROUNDBACKGROUND

Vacancy chains:Vacancy chains:Given the vacancy transition matrix Given the vacancy transition matrix PP,,

P P ==

Suppose Suppose ss33 is the only absorbing state, then is the only absorbing state, then

==

and…and…

BACKGROUNDBACKGROUND

is the vacancy chain.is the vacancy chain.

An entry of the matrix represents the expected An entry of the matrix represents the expected number of times that a vacancy emerging from state number of times that a vacancy emerging from state ii will appear in state will appear in state jj before it is absorbed. before it is absorbed.

RESULTSRESULTS

Proposition 1Proposition 1: : For any two triangular vacancy For any two triangular vacancy

transition matrices transition matrices QQ and and PP, if , if QQ is more mobile than is more mobile than PP

(i.e., Q=PR), then(i.e., Q=PR), then for all i for all i. .

RESULTSRESULTS

Example:Example:Suppose thatSuppose that

PP = = QQ = =

such that such that QQ==PRPR

RR = = and thus and thus QQ is more is more

mobile then mobile then PP..

RESULTSRESULTS

PP ==

==

RESULTSRESULTS

QQ ==

==

RESULTSRESULTS

RESULTSRESULTS

Proposition 2Proposition 2: : For doubly stochastic transition For doubly stochastic transition

matrices P and Q, if Q is more mobile than P (i.e., 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 Q=PR) and R is doubly stochastic, then the sum of the

entries in the column of vacancy chain of RP is greater entries in the column of vacancy chain of RP is greater

or equal to the sum of the entries in the respective or equal to the sum of the entries in the respective

column of the vacancy chain of Q.column of the vacancy chain of Q.

RESULTSRESULTS

Proposition 3Proposition 3:: If If is a strictly row diagonally is a strictly row diagonally

dominant matrix (i.e., dominant matrix (i.e., ), then ), then

the vacancy chain matrix is a strictly diagonally the vacancy chain matrix is a strictly diagonally

dominant of its column entries (i.e., dominant of its column entries (i.e., for all i and for all i and

j).j).

RESULTSRESULTS

CorollaryCorollary: : If If for all i (i.e., any vacancy that for all i (i.e., any vacancy that

emerges in status i is always associated with a change emerges in status i is always associated with a change

of status), then is strictly diagonally dominant in its of status), then is strictly diagonally dominant in its

columns (that is, columns (that is, for all for all ii and and j).j).

RESULTSRESULTS

Proposition 4Proposition 4: For any two triangular transition : For any two triangular transition matrices Q and P, if Q is more mobile than P (i.e., matrices Q and P, if Q is more mobile than P (i.e., Q=PR), thenQ=PR), then . .

Following Following Conlisk and Sommers (1979), Shorrocks Conlisk and Sommers (1979), Shorrocks (1978), and McFarland (1981),(1978), and McFarland (1981), a greater second a greater second largest eigenvalue is associated with a greater largest eigenvalue is associated with a greater speed of convergence of a transition matrix to its speed of convergence of a transition matrix to its equilibrium (i.e., to a constant row matrix).equilibrium (i.e., to a constant row matrix).

RESULTSRESULTS

Proposition 5Proposition 5:: For any normal transition matrices For any normal transition matrices PP and and

QQ and a doubly stochastic matrix and a doubly stochastic matrix RR, , if if QQ is more mobile is more mobile

than than PP (i.e., (i.e., QQ==PRPR), then ), then . .

Consistent with Parker and Rougier (2001) mobility measure. Consistent with Parker and Rougier (2001) mobility measure.

SummarySummary

We develop a link between the literature on mobility We develop a link between the literature on mobility measures and the vacancy chain literature;measures and the vacancy chain literature;

We derive implications of the mobility measure We derive implications of the mobility measure Q=PRQ=PR for for vacancy chains;vacancy chains;

THE ENDTHE END