The Effect of Expected Income on Individual Migration Decisions John Kennan and James R. Walker1 University of Wisconsin-M adison and NBER July 2001 1 Department of Economics, University of Wisconsin, 1180 Observatory Drive, Madison, WI 53706; jkennan@ssc.wisc.edu and walker@ssc.wisc.edu. The National Science Foundation (Kennan) and the NICHD (Walker) provided research support. We are grateful to Ken Wolpin, Michael Keane, Derek Neal, Peter Arcidiacono, Phil Haile, Jim Ziliak and seminar and conference participants at Yale, Iowa, Duke, Rochester and Wisconsin for helpful comments. 1 Introduction Migration is a complex process that has been studied in several branches of the social sciences. Even within the economics literature, the number of conjectured determinants and the range of consequences considered is impressively diverse (see Greenwood [1997] and Lucas [1997] for recent surveys). In this paper we focus on expected income as the main economic influence on migration. We build an economic model of individual migration decisions, and estimate it on panel data (the NLSY79). Estimating a structural dynamic model of the destination-specific streams has apparently not been done before, perhaps because the required computations have not been feasible. 2 Our basic empirical question is the extent to which people move for the purpose of improving their income prospects. Work by Keane and Wolpin (1997) and by Neal (1999) indicates that individuals make surprisingly sophisticated calculations regarding schooling and occupational choices. Given the magnitude of geographical wage differentials, and given the findings of Topel (1986) and Blanchard and Katz (1992) regarding the responsiveness of migration flows to local labor market conditions, we would expect to find that income differentials play an important role in individual migration decisions. 3 We model individual decisions to migrate as a job search problem in which welfare benefits or other 4 alternative sources of income act as a floor, insuring workers against bad job search outcomes. A worker can draw a wage only by visiting a location, thereby incurring a moving cost. Locations are distinguished by known differences in mean wages, amenity values and alternative income sources. A worker starts the life-cycle in some home location and must determine the optimal sequence of moves before settling down. There is a two-dimensional ranking of locations, ex ante: some places have high mean wages, and others have attractive fallback options (both adjusted for amenity values). In addition we allow for a bias in favor of the home location. The decision problem is too complicated to be solved analytically, so we use a discrete approximation that can be solved numerically, following Rust (1994). The parameters of the model include a discount 2 Holt (1996) estimated a dynamic discrete choice model of migration, but his framework modeled the move/stay decision and not the state-specific flows. 3 Blanchard and Katz (1992, p.2), using average hourly earnings of production workers in manufacturing, by state, from the BLS establishment survey, describe a pattern of “ strong but quite gradual convergence of state relative wages over the last 40 years.” For example, using a univariate AR(4) model with annual data, they find that the half-life of a unit shock to the relative wage is more than 10 years. Similar findings were reported by Barro and Sala-i-Martin (1991) and by Topel (1986). 4 This differs from the standard job search model in which unemployment benefits are treated as a subsidy received while search continues. In our model, welfare provides a safety net in case the search fails. 1 factor, a risk aversion coefficient and a home premium summarizing individual preferences; moving costs, including a fixed cost and a cost that is proportional to distance; means and variances of wages in each location; a relative variance parameter governing the extent to which individual wage offers are correlated across locations and a persistence parameter governing the relative importance of permanent and transitory components of wages. We also allow for differences in location size, measured by the population in origin and destination locations. 2 An Optimal Search Model of Migration We model migration as an optimal search process. The basic assumption is that wages are local prices of individual skill bundles. The individual knows the wage in the current location, but not in other locations, and in order to determine the wage at each location, it is necessary to move there, at some cost. In each location there is also a fallback option, such as welfare or family support, and the value of this is known. The model aims to describe the migration decisions of young workers in a stationary environment. The wage offer in each location may be interpreted as the best offer available in that location. Wages are permanent, so the only chance of getting a better offer is to move to a new location. It may be that wage differentials across locations equalize amenity differences, but a stationary equilibrium with heterogeneous worker preferences and skills still requires migration to redistribute workers from where they happen to be born to their equilibrium location. Alternatively, it may be that wage differentials are slow to adjust to location-specific shocks, because gradual adjustment is less costly for workers and employers. In that case, our model can be viewed as an approximation in which workers take current wage levels as a rough estimate of the wages they will face for the foreseeable future. In any case, the model is intended to describe the partial equilibrium response of labor supply to wage differences across locations; from the worker’s point of view the source of these differences is immaterial, provided that the differences are permanent. A complete equilibrium analysis would of course be much more difficult, but our model can be viewed as a building-block toward such an analysis. Suppose there are J locations, and individual i’s wage W ij in location j is is a random variable with distribution function F ij . The fallback option is bj , and thus income in location j is yij = max [W ij ,bj ]. Migration decisions are made so as to maximize the expected discounted value of lifetime utility, subject to budget constraints. Consider a person with “ home” location h, who is in location R this period and in location j next period. The flow of utility in the current period for such a person is specified as 2 1& γ uh (C; R, j) ' α C & 1 % a % κχ R { j' h } & δ0 χ{ j…R} & δ1 M(R,j) 1&γ The notation is as follows. C is consumption in the current period and γ $ 0 is a constant relative risk aversion coefficient. The value of amenities is aR, and there is a premium κ that allows each individual to have a preference for their native location (χA is used as an indicator meaning that the statement A is true). The disutility associated with moving is an affine function of the distance M(R,j) from origin to destination. In general, the level of assets is an important state variable for this problem, but we will focus our analysis on a special case in which assets do not affect migration decisions. Suppose the marginal utility of income is constant (γ = 0 in the specification above), and individuals can borrow and lend without restriction at a given interest rate. Then expected utility maximization reduces to maximization of expected lifetime income, net of moving costs, with the understanding that the value of amenities is included in income, and that both amenity values and moving costs are measured in consumption units. 5 This is a natural benchmark model, although of course it imposes strong assumptions. 6 There is little hope of solving this problem analytically. In particular, the Gittins index solution of the 7 multiarmed bandit problem cannot be applied because there is a cost of moving. But by using a discrete approximation of the wage distribution in each location, we can compute the value function and the optimal decision rule by standard dynamic programming methods, following Rust (1994). Let F j be the wage distribution function in location j. We approximate this by a discrete distribution j &1 s & 1 , where s = 1,2,...,n. Then F is approximated by a over n points, as follows. Let as ' Fj j n 2n uniform distribution over the set {ajs}n . For example, if n = 10, the approximation puts probability 1/10 s=1 th th th on the 5 , 15 , ... 95 percentiles of the distribution F. 5 Note that this neatly sidesteps the question of whether moving costs should be specified as “ psychic” costs that directly reduce utility, or as monetary costs that reduce disposable income. With constant marginal utility of income, there is no meaningful difference between these two specifications. 6 Even if the marginal utility of consumption is not constant, one can still compute the increase in current-period consumption needed to just offset the utility cost of moving, and use this to translate the utility cost into an income equivalent. Then the optimal migration problem can be viewed as maximization of net lifetime income, and this will be a good approximation if the compensating variation in consumption is roughly constant. But this argument rests on the assumption that the individual can borrow against future income (including income generated by a move) in order to sustain current consumption. 7 See Banks and Sundaram (1994) for an analysis of the Gittins index in the presence of moving costs. 3 2.1 The Value Function Consider a person currently in location R, with a J-vector ω summarizing what is known about wages in all locations. Here ωj is either 0 or an integer between 1 and n, with the interpretation that if ωj = s > 0, j then the wage in location j is known to be as, and if ωj = 0 then the wage in location j is still unknown, so j that if the person moves to j, the wage will be as with probability 1/n, for 1 # s # n. The value function for a native of location h can be written in recursive form as 1 n Σ V ( R,ω1,ÿ,ωR&1, s, ωR%1,ÿ,ωJ ) if ωR ' 0 n s'1 h V h(R,ω) ' max uh yR(ωR )& ∆(R ,j ); R ,j % β Vh( j,ω) if ωR > 0 j where yj (ω) = max[bj ,aωj ], and ∆(R,j) is the monetary cost of moving from R to j. We compute Vh by value function iteration. It is convenient to use Vh(R,ω) / 0 as the initial estimate, so that if T is the number of iterations, the result gives the optimal policy for a (rolling) T-period horizon. 3 Empirical Implementation An important limitation of the discrete dynamic programming method is that the number of states is typically large, even if the search problem is relatively simple. If there are J locations and the discrete J approximation of the wage distribution has n points of support, the number of states is J(n+1) . For example a model with J=5 and n=10 has 805,255 states. Although value functions for such a model can be computed in a few hours, estimation of the structural parameters requires that the value function be computed many times. Estimation becomes infeasible unless the number of structural parameters is small. Ideally, locations would be defined as local labor markets. T he smallest geographical unit identified in the NLSY is the county, but we obviously cannot let J be the number of counties, since there are over 3,100 counties in the U.S. Indeed, even restricting J to the number of states still far exceeds current computational capabilities. T o aggregate locations beyond the state level (e.g. Census Regions) is uninterpretable; for example, we lose the ability to identify the effects of state benefit systems. Consequently, we define locations as states, but restrict the information available to each individual, as explained below. 4 3.1 Outline of the Estimation Method We first expand the model to allow for unobserved heterogeneity in individual payoffs. Let ζ = (ζ1,ζ2,...,ζJ) be a vector of idiosyncratic utility adjustments that are known to the worker before the migration decision is made in each period (but not observed by the econometrician). We assume that each component ζj is drawn independently according to a distribution function π; also, these draws are independent across individuals and over time. The individual’s value function is then given by n Σ p s Vh ( R,ω1 ,ÿ,ωR&1 , s, ωR%1 ,ÿ,ωJ ,ζ;θ) if ωR ' 0 s' 1 Vh(R,ω,ζ;θ) ' max u h yR( ωR )&∆(R , j) ;R , j ¯ % ζ j % β Vh( j,ω;θ) if ωR > 0 j ¯ where θ is the vector of unknown parameters and the expected value function V is defined by ¯ V(j,ω; θ) / IV(j,ω,ζ;θ)dπ(ζ) If π is the Type 1 Extreme Value distribution8 then, using arguments due to McFadden (1973) and Rust ¯ (1987) we can show that the function V satisfies n ¯ Σ ps Vh (R,ω1,ÿ,ωR&1,s,ωR%1,ÿ,ωJ ; θ) if ωR ' 0 s'1 ¯ Vh(R,ω; θ) ' J log Σ exp [vhj (R,ω; θ) ] if ωR > 0 j'1 where ¯ vhj (R,ω;θ) ' uh yR(ωR )&∆(R,j ); R ,j % β V h(j,ω;θ) This gives the probability, Pr(d(j)=1 | h, R,ω), that a native of h in location R with information ω will move to location j: 8 A random variable X is exponentially distributed if exp(-X) is uniformly distributed on [0,1]. Repeating this yields an extreme value distribution: Y has the extreme value distribution if exp(-Y) is exponentially distributed. 5 exp [vhj (R,ω;θ)] Pr(d(j)'1*h,R,ω; θ) ' J Σ exp [vhτ(R,ω; θ)] τ'1 3.2 A Limited History Approximation When the number of locations is moderately large, the model becomes computationally infeasible. 9 This is a common problem with discrete dynamic programming models, and various devices have been proposed to deal with it. In our context it seems natural to use an approximation that takes advantage of the timing of migration decisions. So far, we have assumed that information on the value of human capital in alternative locations is permanent, and so if a location has been visited previously, the wage in that location is known, no matter how much time has passed. This means that the number of possible states increases exponentially with the number of locations visited. In practice, however, the number of people seen in many distinct locations is very small. Thus by restricting the information set to include only wages seen in recent locations, it is possible to drastically shrink the state space while retaining most of the information actually seen in the data. Specifically, we suppose that the number of wage observations cannot exceed M, with M < J, so that it is not possible to be fully informed about wages at all locations. M Then if the wage distribution in each of J locations has n points of support, the number of states is (Jn) , since this is the number of possible M-period histories describing the locations visited most recently, and the wages found there. For example, if J is 50 and n is 5 and M is 2, the number of states is 62,500, which is manageable. This approximation reduces the number of states in the most obvious way: we simply delete most of them. Someone who has “ too much” wage information in the big state space is reassigned to a less- informed state. Individuals makes the same calculations as before when deciding what to do next, and the econometrician uses the same procedure to recover the parameters governing the individual's decisions. There is just a shorter list of states, so two people who have different histories may be in different states in the big model, but they are considered to be in the same state in the reduced model. In particular, two people who have the same recent history are in the same state, even if their previous histories were 9 And it will remain so, even if computers improve: for example, if a location is a State, and the wage distribution has 5 support points, then the number of dynamic programming states is 40,414,063,873,238,203,032,156,980,022,826,814,668,800. 6 different (and two people who have different wage information now may have the same wage information following a move). In order to compute the likelihood function using this approximation, it is convenient to redefine 0 1 M-1 notation. Let R = (R ,R ,...R ) be an M-vector containing the sequence of recent locations (beginning with the current location), and let ω be the corresponding sequence containing recent wage information. Then the probability that an individual in state (R,ω) will move to location j can again be written in the form exp [vhj (R,ω;θ)] Pr(d(j)'1*h,R,ω; θ) ' J Σ exp [vhτ(R,ω; θ)] τ'1 where vj is now defined as vhj(R,ω; θ) ' uh yR0(ωR0 ) & ∆(R0, j) ¯ % β V h ( j,R0,R1,ÿ,RM&2 ),( ωj,ω0,ω1,ÿ,ωM&2 );θ with n 1 Σ V (R,( s,ω1,ω2,ÿ,ωM&1); θ) if ω0 ' 0 ¯ n s'1 h ¯ Vh(R,ω; θ) ' J log Σ exp [vhj (R,ω;θ)] if ω0 > 0 j'1 3.3 Population Effects It has long been recognized that location size matters in migration models. 10 California and Wyoming cannot reasonably be regarded as just two alternative places, to be treated symmetrically as origin and destination locations. To take one example, a person who moves to be close to a friend or relative is more likely to have friends or relatives in California than in Wyoming. A convenient way to model this in our framework is to allow for more than one draw from the distribution of preference shocks in each location. Specifically, we assume that the number of draws per location is an affine function of the number of people already in that location, and that migration decisions are controlled by the maximal draw for each location. This leads to the following modification of the logit function describing migration probabilities: 10 See T. Paul Schultz (1982). 7 (1% δ3 nj ) exp [v hj (R,ω; θ)] Pr(d(j)'1*h,R,ω; θ) ' J Σ (1% δ3 nτ ) exp [vhτ (R,ω;θ)] τ'1 Here nj denotes the population in location j, and the (nonnegative) parameter δ3 can be interpreted as the number of additional draws per person. 3.4 Computation Since the parameters are embedded in the value function, computation of the gradient and hessian of the loglikelihood function is not a simple matter (although in principle these derivatives can be computed in a straightforward way using the same iterative procedure that computes the value function itself). We maximize the likelihood using an “ amoeba” algorithm that implements the downhill simplex method of Nelder and Mead. T his method does not use derivatives, and it seems appropriate for problems such as this in which there is no reason to expect that the loglikelihood function is concave. In practice the method works well for the models we have estimated so far; in particular, it is robust to large changes in the starting values of the parameters. On the other hand, the method is slow, and so we also use gradient methods to speed up the computations, particularly when doing sensitivity analysis.11 4 Migration and Welfare We analyze the migration decisions of low income women at risk to receive AFDC. This is a natural application of our model, because location-specific benefits in the model are most directly related to welfare benefits (AFDC and Food Stamps) within each state. The recent literature on welfare-induced migration is summarized by Meyer (1999). While the consensus view from earlier work reviewed by Moffitt (1992) was that differences in welfare benefits across states had a significant effect on migration decisions, subsequent studies by Levine and Zimmerman (1995) and by Walker (1994) found little or no effect. Meyer argued that by paying careful attention to the determinants of welfare participation, the ambiguity in these results can be resolved in favor of a significant (but small) effect of welfare on migration. Gelbach (2000) also found a significant effect, arguing that previous studies had failed to properly account for dynamic selection effects. None of 11 An example of our (fortran) computer program can be found at www.ssc.wisc.edu/~jkennan/research/mbr8.f90. 8 these studies contains a complete dynamic choice model, however, and we believe that our model can provide a more systematic analysis. 4.1 Earnings and Welfare Benefits We restrict the estimation sample to women from the non-military subsample of the NLSY79 with twelve or fewer years of education. The observational window begins in the year the woman is first single with a dependent child. To be included in the sample, information on residence must be observed for at least two periods. We follow these respondents either until the end of their single parenthood, the end of the sample period (1992) or the first wave in which they are not interviewed. There were 1,728 people satisfying these restrictions, and we have data on 10,101 location decisions (i.e. person-years). For each respondent, the wage in each State is measured as the sum of annual wage and salary income divided by total weeks worked, for all years residing in that State. We use PUMS data from the 1990 Census to estimate wage distributions for each State. Benefits correspond to the combined AFDC and Food Stamp benefits for a family of 3 in 1989. Appendix Table 1 reports parameter values used in estimation. Table 1 shows that the differences in benefits across states are large: for example the highest annual benefit (excluding Alaska and Hawaii) is $7,568 in California, and the lowest is $3,426, in Alabama (in 12 1983 dollars). In the third column of the table, these differences are adjusted for differences in living costs across states, using the ACCRA cost of living index (http://www.coli.org/). Even after this adjustment, the differences remain large. The last column of the table shows the wage percentile in the 1990 PUMS data corresponding to the benefit level, by State. The typical situation is that less than 50% of single women with children earn more than the benefit level. 12 Using a slightly different definition of benefits, and after adjusting for cost of living differences, Gelbach reports a benefit level of $9,912 in Connecticut, and $4,654 in Mississippi (in 1997 dollars). 9 4.2 Partial Likelihood Estimates We condition on the estimated earnings distributions for each state and maximize the partial likelihood to obtain estimates of the utility function parameters. We fix β at 0.9. The results in Table 2 show that differences in expected income are a significant determinant of migration decisions for this population, but this effect cannot be accurately measured without controlling for other influences on migration. There are 10,101 person-years in the data, and there are 380 interstate moves. This is an annual migration rate of 3.76%, and the first column in Table 2 matches this rate by 1 389 setting the probability of moving to each of J-1 locations to a constant value, namely , with J&1 10,171 J = 51. 13 The second column shows the effect of income with no other variables included, but the next three columns show that population size, distance, and home location all have highly significant effects on migration. 14 The last two columns show the effect of income, controlling for these other effects; the last column uses wage and benefit numbers adjusted for cost of living differences across States, while the previous column uses unadjusted data. One way to interpret the estimated coefficients is to consider what the migration rate would be in the absence of some of the effects measured in Table 2. Suppose for example that there were no differences in population or income across States, and that there were no distance between States. Then the migration rate would be much higher: the coefficient estimates in the last column of Table 2 imply an annual migration probability of 16.8% for someone currently in the home location, and a probability of 23.2% for someone who is currently in a different location. The estimated effect of expected income is quantitatively important. For example, a permanent increase in income of $4,000 per year in the current location reduces the annual migration probability from 3.82% to 2.00%, while an income reduction of $4,000 increases the migration probability to 7.19%. Thus large differences in expected income lead to large migration flows (since these probability differences are cumulated over time). How Big are the Moving Costs? &δ0 e ' 1 13 389 ; the solution is In other words the estimate of δ0 solves the equation 1%(J&1)e &δ0 J&1 10,171 δ0 = log(489100) - log(389). 14 The χ2 statistics in the table are for likelihood ratio tests of the form 2log(LU/LR) ~ χ²(r), where r is the number of restrictions embodied in LR relative to LU. 10 Since the utility function is linear in income, we can translate the estimated moving cost into a dollar equivalent. The result of this calculation is δ0/α = $306,483, with the interpretation that the (lump-sum) compensation needed to just offset the cost of a move is enormous: other things equal, an offer of $300,000 would not be enough to persuade someone to move. It may seem that this large moving cost is an artifact of the specification of the model. For example, in the absence of any moving cost, allowing preference shocks to be drawn randomly over J locations implies a migration probability of (J-1)/J, so that with J = 51, nearly everybody moves every period. The first column of Table 2 shows how large the fixed cost of moving has to be in relation to the preference shocks, in order to reduce the migration rate from 50/51 to the observed rate of 3.82%, when all other influences on migration are suppressed. The second column shows that the migration cost must be increased slightly when the influence of income is taken into account – income differences promote some moves and retard others, and on balance the effect is evidently in favor of an increase in migration. More importantly, the estimated moving cost is very large in relation to the income coefficient. When the effects of population and distance and the home premium are taken into account, the estimated moving cost is smaller in relation to income, but it is still very large. To understand why the estimated moving cost is so big, it is helpful to consider an example in which income differentials and moving costs are the only influences on migration decisions. Suppose there are just two income levels, and let y be the present value of the difference between the two incomes. Then it is straightforward to show that the probability of staying in the current location is δ0 e λL ' δ e 0 %J L&1 %J H e α y (11) e δ0 λH ' δ e 0 %J H&1 %J L e &α y where λL is the probability of staying in a low-income location and JL is the number of such locations, and similarly for λH and JH. Given data on λL and λH, these equations can be solved for δ0 and αy, and the ratio C = δ0/αy represents the moving cost as a multiple of the income differential. Note that if C # 1 then 1 λL # , implying a migration rate that is much higher than the rate seen in the data. On the other hand 1%J H if C > 1, then the migration rate can be made arbitrarily small, by increasing both δ0 and α (in fixed proportion). In other words, as long as the moving cost exceeds the income differential (even by a tiny 11 amount) the model can fit a low migration rate by scaling up both the moving cost and the marginal utility of income (which is the same thing as scaling down the preference shocks). Thus if income differentials and the moving cost are both large relative to the preference shocks, the overall migration rate may be low even though the moving cost is not much bigger than the income differential. But an implication of this scenario is that the migration rate from low-income locations should be much larger than the migration rate from high-income locations, and this is evidently at odds with the data. The relationship between observed migration rates and estimated moving costs in this simplified model is illustrated in Figure 1. It is assumed that the average migration rate matches the data, meaning λL %λH that ' 1& 389 . The equations for λL and λH are solved for δ0 and αy, and the implied value of C 2 10,171 is plotted against ψ / λH - λL . The result is that if λL and λH, are close, then the moving cost must be a large multiple of the income differential. So this example suggests that the large estimated moving cost implied by the estimates in Table 2 arises because the empirical relationship between migration rates and income levels is relatively weak. There are of course important influences on migration decisions that are not included in our model, and a reasonable interpretation of the results is that, on average, the omitted variables strongly favor staying in the current location. If this is so, a more complete model would yield a more plausible estimate of the moving cost. Nevertheless, our estimate of the effect of income differentials is valid provided that the variation in the omitted variables across States is not correlated with the income differentials. Conclus ion We have developed a tractable econometric model of optimal migration in response to income differentials across locations. Empirical results show a significant effect of income differentials on migration, for unskilled single women with dependent children who are eligible for AFDC. References 12 Barro, Robert J. and Xavier Sala-i-Martin, “ Convergence across States and Regions,” Brookings Papers on Economic Activity, 1991, 1, 107-158. Blanchard, Olivier Jean and Lawrence F. Katz, “ Regional Evolutions,” Brookings Papers on Economic Activity, 1992, 1, 1-37. Greenwood, Michael J. “ Internal Migration in Developed Countries,” in Handbook in Population and Family Economics Vol. 1B, edited by Mark R. Rosenzweig and Oded Stark. New York: North Holland. 1997. Holt, Frederick (1996) “ Family Migration Decisions: A Dynamic Analysis,” unpublished paper, University of Virginia. Keane, Michael P. and Kenneth I. Wolpin, “ The Career Decisions of Young Men,” Journal of Political Economy, 105(3), June 1997, 473-522. Levine, Phillip B., and David J. Zimmerman “ An Empirical Analysis of the Welfare Magnet Debate Using the NLSY,” Journal of Population Economics 2000. Lucas, Robert E. B. “ Internal Migration in Developing Countries,” in Handbook in Population and Family Economics Vol. 1B, edited by Mark R. Rosenzweig and Oded Stark. New York: North Holland. 1997. Meyer, Bruce D. “ Do the Poor Move to Receive Higher Welfare Benefits?” Northwestern University, October 1999. Moffitt, Robert (1992), "Incentive Effects of the United States Welfare System: A Review," Journal of Economic Literature 30 (March): 1-61. Neal, Derek, "The Complexity of Job Mobility of Young Men," Journal of Labor Economics, April, 1999, 237-261. Rust, John (1987) “ Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher,” Econometrica, 55 (5), September 1987, 999-1033. Rust, John (1994), “ Structural Estimation of Markov Decision Processes,”in Handbook of Econometrics, Volume IV. Edited by Robert F. Engle and Daniel L. McFadden. New York: Elsevier. Rust, John (1996), "Numerical Dynamic Programming," in Handbook of Computational Economics, North Holland, H. Amman, D. Kenrick, and J. Rust eds. Schultz, T. Paul (1982) “ Lifetime Migration within Educational Strata in Venezula: Estimates of a Logistic Model,” Economic Development and Cultural Change 30: 559-593. Topel, Robert H., “ Local Labor Markets,” Journal of Political Economy, 1986, 94(3), part 2, S111-S143. Walker, James R. (1994) "Migration Among Low-Income Households: Helping the Witch Doctors Reach Consensus," Discussion Paper, #94-1032, Institute For Research on Poverty. 13 Table 0: Interstate Migration Flows, NLSY79 Less than High High School Some College College School No. of people 1768 3534 1517 1435 Horizon (years) 5 13 5 13 5 13 5 13 No. of movers 334 423 598 771 327 376 441 469 Repeat moves 239 434 313 653 167 264 196 261 Repeat moves as % of all moves 41.7 50.6 34.4 45.9 33.8 41.3 44.4 35.7 Movers (%) 18.9 23.9 16.9 21.8 21.6 24.8 30.8 32.7 Moves Per Mover 1.7 2.0 1.5 1.8 1.5 1.7 1.4 1.6 Return Migration ( % of all moves) Return - Home 22.9 24.0 20.6 24.1 16.0 17.5 12.4 13.4 Return - Else 5.4 12.4 2.7 7.2 2.8 5.9 2.5 3.3 Movers who return home (%) 39.2 48.7 31.4 44.5 24.2 29.8 17.9 20.9 Return-Home: % of Repeat 54.8 47.5 60.1 52.5 47.3 42.4 40.3 37.5 14 Table 1: Wages and Benefits, by State Single Women with Children, 1989 Observations, NLSY Benefits Adjusted Wage (Person-years) Benefits Percentile Alabama 385 3,426 3,604 55.6% Alaska 67 9,765 7,232 73.9% Arizona 81 5,061 4,894 56.7% Arkansas 207 4,258 4,517 63.4% California 1,066 7,568 6,877 70.6% Colorado 129 5,667 5,667 57.5% Connecticut 910 259 7,297 5,948 67.0% Delaware 10 5,332 4,972 59.6% DC 102 5,739 4,625 55.4% Florida 394 5,023 4,954 53.2% Georgia 648 4,897 5,036 59.7% Hawaii 5 8,381 6,325 66.7% Idaho 3 5,139 5,291 64.3% Illinois 311 5,448 5,316 61.0% Indiana 116 5,032 5,228 50.7% Iowa 31 5,748 5,879 62.6% Kansas 75 6,126 6,353 68.4% Kentucky 56 4,394 4,665 63.1% Louisiana 134 4,123 4,136 72.2% Maine 129 6,048 6,307 68.3% Maryland 193 5,806 5,729 52.1% M assachusetts 425 6,735 5,874 64.7% Michigan 394 6,774 6,485 69.6% Minnesota 107 6,687 6,643 61.5% Mississippi 165 3,445 3,601 62.6% Missouri 336 5,013 5,365 59.2% Montana 72 5,516 5,524 66.0% Nebraska 27 5,545 5,921 46.5% Nevada 5 5,313 4,910 54.8% New Hampshire 0 6,445 5,313 46.3% New Jersey 284 6,029 5,178 53.3% New Mexico 81 4,839 4,807 60.1% New York 546 6,890 6,342 67.7% North Carolina 405 4,858 4,918 50.9% North Dakota 0 5,700 5,840 80.0% Ohio 621 5,294 5,343 63.6% Oklahoma 83 5,284 5,331 58.9% Oregon 37 6,271 6,092 66.9% Pennsylvania 323 5,806 5,705 60.1% Rhode Island 12 6,629 5,988 73.4% South Carolina 414 4,277 4,405 48.6% South Dakota 15 5,565 5,803 62.5% Tennessee 174 3,958 4,196 52.5% Texas 728 4,065 4,113 57.8% Utah 26 5,632 5,807 57.5% Vermont 39 7,345 6,768 71.4% Virginia 229 5,477 5,553 56.6% Washington 532 123 6,552 6,554 63.8% West Virginia 74 4,694 4,788 67.9% Wisconsin 348 6,581 6,860 66.4% Wyoming 1 5,516 5,491 58.6% All States 10,101 5,670 5,479 Table 2: Interstate Migration of Unskilled Single Women with Dependent Children Disutility of Moving 7.1650 7.2086 7.0036 6.2652 5.0441 5.0938 5.1090 0.0528 0.0569 0.0529 0.0823 0.0813 0.0847 0.0805 Population 0.1067 0.1110 0.1316 0.1234 0.1243 0.0088 0.0086 0.0129 0.0129 0.0128 Distance 0.8044 0.7133 0.7108 0.7090 (1000 miles) 0.0824 0.0802 0.0802 0.0805 Home Premium ---------- 0.3822 0.3874 0.3873 0.0143 0.0144 0.0144 Income(/$10K) 0.0725 ---------- ---------- 0.1315 0.0273 0.0326 “ Real” Income (ACCRA) ---------- ---------- ---------- 0.1633 0.0331 Loglikelihood -3044.494 -3040.368 -2975.605 -2929.924 -2684.209 -2676.910 -2674.822 2 χ (1) 8.252 129.778 91.362 491.430 14.598 18.774 p-value 0.0041 0.0001 0.00001 Notes: Estimated asymptotic standard errors are given in italics below the coefficient estimates. The length of the horizon is 40 years, with discount factor β = .9 The wage distributions have 3 points of support Appendix Table 1: Population Size and Empirical Approximation of Income Distribution By State(in 1983 Dollars) Cut point P roportion Income Value Income Value between Cells State P opulation Censored Benefits Cell 2 Cell 3 2 and 3 Alabama 3893888 .5560 3426 6048 11290 8452 Alaska 401851 .7391 9765 12903 17339 13548 Arizona 2718215 .5669 5061 8065 15323 10484 Arkansas 2286435 .6340 4258 7132 12215 9677 California 23667902 .7059 7568 10081 17742 13558 Colorado 2889964 .5746 5497 8065 14516 10452 Connecticut 3107576 .6695 7294 11694 19355 10081 DC 638333 .5536 5739 10484 16935 13710 Florida 9746324 .5324 5023 8065 14113 10081 Georgia 5463105 .5969 4897 7258 13710 10484 Hawaii 964691 .6667 8381 10484 15323 12860 Ido 943935 .6429 5139 6956 11290 8065 Illinois 11426518 .6103 5448 8468 15323 11290 Indiana 5490224 .5068 5032 8065 13710 10214 Iowa 2913808 .6264 5748 8008 14516 10081 Kansas 2363679 .6842 6126 7963 13145 9677 Kentucky 3660777 .6310 4394 7032 14253 10323 Louisiana 4205900 .7224 4123 6202 11129 8165 Maine 1124660 .6829 6048 8402 13710 12097 Maryland 4216975 .5208 5806 9677 17742 12097 Massachusetts 5737037 .6474 6735 10161 17742 13306 Michigan 9262078 .6955 6774 9677 18548 12093 Minnesota 4075970 .6150 6687 8871 16129 12097 Mississippi 2520638 .6261 3445 4919 9677 6452 Missouri 4916686 .5919 5013 7279 12903 9677 Montana 786690 .6596 5516 8380 13520 9677 Nebraska 1569825 .4648 5545 7661 12903 12016 Nevada 800493 .5476 5313 8065 16129 11290 New Hampshire 920610 .4630 6445 12903 21774 16129 Population Size and Empirical Approximation of Income Distributions By State (continued) Cut point P roportion Value 1 between Cells State P opulation Censored (Benefits) Value 2 Value 3 2 and 3 New Jersey 7364823 .5333 6029 9677 17432 13710 New Mexico 1302894 .6011 4839 7258 12097 8871 New York 17558072 .6771 6890 9677 17742 13242 North Carolina 5881766 .5087 4858 7558 12903 9677 North Dakota 652717 .8000 5700 6876 8871 8085 Ohio 10797630 .6355 5294 8871 16129 12594 Oklahoma 3025290 .5887 5284 6894 12097 9677 Oregon 2619399 .6685 6271 8076 15323 10484 P ennsylvania 11863895 .6009 5806 8856 14516 11290 Rhode Island 947154 .7342 6629 8548 16129 12097 South Carolina 3121820 .4860 4277 7258 12431 9677 South Dakota 690768 .6250 5565 6713 11290 8131 Tennessee 4580367 .5251 3958 7089 12903 9677 Texas 14229191 .5777 4065 6452 12903 9258 Utah 1461037 .5747 5632 8065 14516 10484 Vermont 511456 .7143 7345 10349 17889 14919 Virginia 5346818 .5660 5477 8065 14516 10484 Washington 4132156 .6376 6552 8871 17889 12903 West Virginia 1921005 .6790 4964 6984 13731 9806 Wisconsin 4705767 .6644 6581 9677 16197 11631 Wyoming 469557 .5862 5516 7447 15121 10094