Tion group g in state j. Markov models usually assume time-invariant probabilities (P) of moving between states. However, if individuals both react to and transform features of their neighborhoods through their mobility behavior, then their behavior follows an interactive Markov model (IM) (Conlisk 1976), where the elements of P depend on the population distribution at time t:(8.4)Here m[t] represents the distribution of blacks and Caspase-3 Inhibitor web whites across neighborhoods, and the probability of moving into a given neighborhood is a function of its ethnic composition. In this model, preferences for neighborhood characteristics are fixed, but the attractiveness of specific neighborhoods changes as a result of their changing characteristics.Sociol Methodol. Author manuscript; available in PMC 2013 March 08.Bruch and MarePageTo illustrate the interactive Markov model, we consider a simple city with 2 neighborhoods and a population of 10 blacks and 10 whites. At time 0 the population is completely segregated; all blacks are in one state, and all whites are in the other. Thus, our starting population at time 0 is. Next, we compute the population trajectory for whites and blacks using their respective preference functions. For example, if people evaluate their neighborhoods according to a simplified version of Equation 3.4, where the probability that the ith person selects the jth neighborhood is proportion own-group, then , where Zj is neighborhoodNIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscript(8.5)At the next step 2,(8.6)Sociol Methodol. Author manuscript; available in PMC 2013 March 08.Bruch and MarePageThe process can continue until the neighborhoods reach equilibrium, that is, where m[t + 1] = Pt(m[t]) and m[] = P = Ptm[t]. Given an estimated discrete choice function that can generate the Pt, it is possible to compute the expected pattern of residential segregation under the mobility regime summarized in mobility matrices Pt using the standard measures of residential segregation (Mare and Bruch 2003). Tuljapurkar, Bruch, and Mare (2010) provide a mathematical analysis of Markov models for segregation and neighborhood change. In principle, an interactive Markov models for mobility between individual neighborhoods can be represented as a fixed rate Markov model of mobility between neighborhood types (e.g., Hermanns 2002). General Equilibrium Models with Price Effects Another strategy for studying neighborhood dynamics is using general equilibrium (GE) models with price effects. Bayer and colleagues (Bayer and McMillan 2005, 2008; Bayer, McMillan, and Rueben 2004) use GE models to examine the relationship between residential choice behavior and neighborhood get Thonzonium (bromide) outcomes. The analysis consists of two parts: (1) estimating a discrete choice model and (2) simulating the expected distribution of individuals in each neighborhood implied by the choice model. GE models assume that observed neighborhoods are in equilibrium, such that each individual had made an optimal choice given the choices of all other individuals. The models can be used to show how a new equilibrium distribution of neighborhoods results from some change in initial conditions or behavior (e.g., assuming that people are indifferent to the racial composition of their neighborhoods or assigning all ethnic groups equal income distributions). The first step is assuming or estimating a discrete choice model for the effects of housing prices, neighborhood race/ethnic composi.Tion group g in state j. Markov models usually assume time-invariant probabilities (P) of moving between states. However, if individuals both react to and transform features of their neighborhoods through their mobility behavior, then their behavior follows an interactive Markov model (IM) (Conlisk 1976), where the elements of P depend on the population distribution at time t:(8.4)Here m[t] represents the distribution of blacks and whites across neighborhoods, and the probability of moving into a given neighborhood is a function of its ethnic composition. In this model, preferences for neighborhood characteristics are fixed, but the attractiveness of specific neighborhoods changes as a result of their changing characteristics.Sociol Methodol. Author manuscript; available in PMC 2013 March 08.Bruch and MarePageTo illustrate the interactive Markov model, we consider a simple city with 2 neighborhoods and a population of 10 blacks and 10 whites. At time 0 the population is completely segregated; all blacks are in one state, and all whites are in the other. Thus, our starting population at time 0 is. Next, we compute the population trajectory for whites and blacks using their respective preference functions. For example, if people evaluate their neighborhoods according to a simplified version of Equation 3.4, where the probability that the ith person selects the jth neighborhood is proportion own-group, then , where Zj is neighborhoodNIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscript(8.5)At the next step 2,(8.6)Sociol Methodol. Author manuscript; available in PMC 2013 March 08.Bruch and MarePageThe process can continue until the neighborhoods reach equilibrium, that is, where m[t + 1] = Pt(m[t]) and m[] = P = Ptm[t]. Given an estimated discrete choice function that can generate the Pt, it is possible to compute the expected pattern of residential segregation under the mobility regime summarized in mobility matrices Pt using the standard measures of residential segregation (Mare and Bruch 2003). Tuljapurkar, Bruch, and Mare (2010) provide a mathematical analysis of Markov models for segregation and neighborhood change. In principle, an interactive Markov models for mobility between individual neighborhoods can be represented as a fixed rate Markov model of mobility between neighborhood types (e.g., Hermanns 2002). General Equilibrium Models with Price Effects Another strategy for studying neighborhood dynamics is using general equilibrium (GE) models with price effects. Bayer and colleagues (Bayer and McMillan 2005, 2008; Bayer, McMillan, and Rueben 2004) use GE models to examine the relationship between residential choice behavior and neighborhood outcomes. The analysis consists of two parts: (1) estimating a discrete choice model and (2) simulating the expected distribution of individuals in each neighborhood implied by the choice model. GE models assume that observed neighborhoods are in equilibrium, such that each individual had made an optimal choice given the choices of all other individuals. The models can be used to show how a new equilibrium distribution of neighborhoods results from some change in initial conditions or behavior (e.g., assuming that people are indifferent to the racial composition of their neighborhoods or assigning all ethnic groups equal income distributions). The first step is assuming or estimating a discrete choice model for the effects of housing prices, neighborhood race/ethnic composi.
