ow variables (sections A1. A3.); 2) state-level average earnings (section A4.) and rents (section

Transcription

1 A Data Appendix This data appendix contains detailed information about: ) the construction of the worker ow variables (sections A. A3.); 2) state-level average earnings (section A4.) and rents (section A5.); 3) the role of immigration (section A6.); 4) the econometric procedure associated with the results of Table 7 of the paper (section A7.). A. Sample Selection Data are from the ve percent samples of the Censuses and from the one percent sample of the 970 Census (Form State Sample). All the measures of gross and net ows and the stock of population that are reported in the paper are computed using a sample of individuals that, at the time of the relevant Census: are between 28 and 60 years of age; are not living in group quarters; are in the labor force but not in the armed forces; if foreign-born, have immigrated to the U.S. at least ve years before the Census year; were not living abroad ve years before the Census year; are not living in either Alaska, Hawaii, or the District of Columbia in the Census year or ve years before the Census year. A2. Construction of Flow Rates In this section I describe how I have constructed the alternative measures of worker ows used in Tables, 2, and 4 of the paper. Raw Measures. Each individual i is observed living in a state k at the time of the Census and in a state j ve years before the Census year. Let g i jk 8 >< if i observed moving from j to k gjk i = >: 0 else be de ned as: : The raw individual migration data g i jk are then aggregated up by state of origin-state of destination pairs to create a matrix of bilateral gross ows among the 48 states:

2 2 F jk = X i g i jk: The ows are then converted into bilateral out ow (o) and in ow (i) rates by dividing by a state s population ve years before the Census year (P ): o jk = F jk P j ; i jk = F jk P k : (A.) Finally, the state-level out ow and in ow rates are obtained by summing over states of destination (out ows) and state of origin (in ows): out ow rate j = X k o jk ; in ow rate k = X j i jk : (A.2) The net ow rate for a state is de ned as the di erence between its gross in ow and out ow rates. Composition e ects. The state-level ow rates are constructed as in the previous paragraph, with the variable eg i jk replacing gi jk : The variable egi jk denotes the residual from a logistic regression of g i jk on a set of dummies for education, age, sector, race, occupation, number of children, marital status, sex, and immigration status (i.e. a dummy for whether the worker is foreign-born or not). Note that the residual eg i jk takes a value in the interval (0; ) for each worker i. Gravity. States might display certain bilateral patterns of in ows and out ows due to their relative size and their distance from one another (the gravity forces ). In order to control for gravity, I rst separately regress the bilateral in ow and out ow rates (de ned in equation A.) on log distance, log distance squared, log population of origin and destination states, and the squares of these two variables, a border dummy, and the interactions between the border dummy and log distance, log population of the origin state, log population of the destination state. The residuals of these two equations for bilateral in ow and out ow rates

3 3 are then aggregated into an out ow and in ow rate for each of the 48 states (as in equation A.2). Again, the net ow rate for a state is de ned as the di erence between the gravityadjusted gross in ow and out ow rate. Composition e ects and gravity. Given the bilateral ows, the procedure is the same as in the previous paragraph (Gravity). In turn, the bilateral ows are obtained following the same procedure as in the second paragraph (Composition e ects). Exclude Outliers. The data are constructed as in the rst paragraph (Raw Measures), but it excludes individuals who are living in Nevada, New York, New Jersey, Connecticut, Massachusetts and Delaware either in the Census year or 5 years before. A3. Demographic Groups In order to compute the within-group correlations in Table 3 of the paper, I construct 385 demographic groups based on the following variables and data from the 2000 Census: Age; 7 age groups: 28-3, 32-36, 37-4, 42-46, 47-5, 52-56, Education; 5 education groups: high-school dropout, high-school diploma, some college, college degree, above college. Industry of employment; industries: () agriculture, shing, forestry, hunting and mining, (2) construction, (3) manufacturing non-durables, (4) manufacturing durables, (5) transportation, communication and other public utilities, (6) wholesale and retail trade, (7) nance, insurance, and real estate, (8) business and repair services, (9) personal services, entertainment and recreation services, (0) professional and related services, () public administration. For each of the 385 demographic groups I then construct out ow, in ow and net ow rates by dividing the worker ows by the group s population living in a given state ve years prior to the Census year. These group-speci c rates are used to compute the statistics in Table 4.

4 4 A4. Weekly Earnings Workers weekly earnings are computed using data from the Census of Population and Housing. The same sample selection criteria listed above in Section A. are also applied in this case. Weekly earnings are obtained by summing, for each worker, annual wage income and business and farm income, and dividing the sum by the number of weeks worked. Data on income and the number of weeks worked refer to the year preceding the Census year. I have dropped from the sample a very small number of observations for which an individual reported zero annual earnings but a positive number of weeks worked. In a few instances, reported earnings by self-employed individuals are negative, and these observations have been dropped as well. Given that earnings refer to the year prior to the Census and the worker s labor force participation status refers to the time of the survey, a small fraction of individuals (about 2.5 percent of the sample) reported zero annual earnings and zero weeks worked in the year prior to the Census. I have also dropped these individuals from the sample. For each Census year, the logarithm of weekly earnings was regressed on the following dummy variables for: workers state of residence in the Census year, a workers age, education, sex and race ( white, black and others ), sector and occupation of employment. The R 2 of these regressions is about 30 percent. The measure of average weekly earnings for each state is represented by the estimates of state xed-e ects in this regression. A5. Rents The Census of Population and Housing provides data on the gross monthly rent paid by a renter. This variable reports the gross monthly rental cost of the housing unit, including contract rent plus additional costs for utilities (water, electricity, gas) and fuels (oil, coal, kerosene, wood, etc.). This information is used to derive a measure of land rents in each U.S. state. Observations on rents are obtained for those workers renting a housing units who

5 5 satisfy the sample selection criteria listed above in Section A.. In each Census year, about 30 percent of the sample obtained from applying those selection criteria rents (as opposed to owns) its housing unit. For example, in the 2000 Census more than one million observations are available for renters. In order to remove the in uence of observable characteristics of a housing unit from the monthly rent, I run a hedonic regression of the logarithm of the rent on state xed e ects and a list of observable characteristics of the housing units. These include dummies for: whether the housing unit is located in a metropolitan area, whether the unit is used commercially, the acreage of the property, the acreage of the house, whether meals are included in the rent, whether the housing unit is in a condominium, whether the housing unit contains a kitchen, the number of rooms, the availability of plumbing facilities, the age of the unit, the number of bedrooms, the number of units in the structure. The measure of average rents at the state level is represented by the estimated state xed-e ects from this regression. A6. Immigration The sample of workers selected according to the criteria spelled out in Section A. of this appendix includes foreign-born workers provided they migrated to the U.S. at least ve years before the Census year. While this selection guarantees that aggregate net ows of workers across U.S. states are zero in each Census year, it also assumes away recent immigration ows, i.e. foreign-born workers who migrated to the U.S. during the ve years prior to a given Census year. Recent immigration might potentially play an important role in a ecting internal migration ows from and into certain states. In order to quantify these e ects and to place the magnitude of internal migration ows in perspective relative to their international counterpart I have used the 2000 Census data to compute for each U.S. state the ratio between the number of recent immigrants that are located in that state in the year 2000 and the state s 995 population. The data indicate that the average rate of recent immigration

6 6 across U.S. states is about.5 percent, while Table indicates that the average in ow rate of workers from the rest of the U.S. in the 2000 Census is about 8. percent. Thus, for the average state internal migration is much larger than recent immigration. Of course, it is well known that recent immigrants tend to cluster in a few locations. There are three states for which the recent immigration rate is at least 30 percent as large as the gross in ow rate due to internal migration: California (47 percent), New York (56 percent), and New Jersey (34 percent). It turns out, however, that for the cross-section of U.S. states there is no signi cant association between rates of recent immigration and both gross and net ow rates due to internal migration. So, there is no evidence supporting the view that states with low gross in ow rates experienced larger than average recent immigration from abroad. This result is consistent with the ndings of Card (200) for U.S. metropolitan areas. A7. Testing the Mechanism: Individual-Level Regression In Section 6 of the paper, I describe the econometric results reported in the second panel of Table 7. The results refer to a logit model for the probability that a worker changes state of residence between the years 995 and The main regressor of interest is a dummy variable taking a value of one if the individual lived in 995 in a di erent state from his/her state of birth and zero otherwise. The set of demographic variables and controls in all the speci cations of the regression are: dummies for education, age, industry, occupation, race, number of children, gender, marital status, foreign-born (except in the regression without immigrants), state of birth (if U.S. born), and state of residence in 995. B Analytical Steps and Non-Stochastic Steady State This appendix contains detailed information on: ) The derivation of the indirect utility function, rent and wage for the benchmark version of the model (section B.); 2) The closedform expressions for the non-stochastic steady state version of the value functions, local

7 7 prices, and expected utility of migration (section B2.). B. Indirect Utility Function, Wage and Rent I describe how to solve out for land and obtain an expression for the static indirect utility and the local wage. Using the model s notation for l c (s; ) (demand for land by consumers) the maximum instantaneous utility can be written as: u (s; ) = w (s; ) r (s; ) l c (s; ) + (l c (s; )) ; (B.) where using the functional form for detailed in Section 4: l c (s; ) = A r (s; ) : (B.2) The local prices are equal to the marginal products of labor and land: w (s; ) = zn l f (s; ) ; r (s; ) = ( ) zn l f (s; ) : (B.3) Combining these two equations gives w (s; ) as a function of r (s; ) and the shock z: w (s; ) = " z ( ) () r (s; ) # : (B.4) In order to eliminate l f (s; ) impose the equilibrium condition in the land market: nl c (s; ) + l f (s; ) = ; and replace in it l c (s; ) from equation (B.2) and l f (s; ) from equation (B.3) to obtain an equation in r (s; ) only:

8 8 A n r (s; ) + n ( ) z r (s; ) = : This equation has a closed form solution for r (s; ) if and only if = ; which is assumed in the benchmark calibration of the model. Under this assumption I obtain: r (s; ) = n A + (( ) z) : (B.5) Replace this expression for r (s; ) into equation (B.4) and rearrange to obtain: w (s; ) = zn + A z ( )! : (B.6) In turn, replacing equations (B.2), (B.5) and (B.6) in equation (B.) and rearranging gives the indirect utility function as function of the state variables z and n only: u (s; ) = n z + A! : (B.7) I have then shown that it is possible to solve out for r (s; ) and land from the original speci cation and rewrite the model as one in which the utility functions take the form (B.7) and the unit wage is given by (B.6). B2. Non-Stochastic Steady State Equilibrium In a non-stochastic steady state, the shock z equals its steady state value of one in all locations, and wages and rents are constant over time and across islands. Each island has a constant population n =. Worker migration might occur only because of the idiosyncratic shocks. The only case in which the behavior of the two groups of workers di ers in the steady state is the one in which b is low enough so that badly matched workers choose to migrate (q b = ), while g is large enough so that well-matched ones do not move (q g = 0).

9 9 Under these circumstances the value functions in equation (5) of the paper simplify to: V g = g u (w; r) + gg V g + ( gg ) (e ) ; (B.8) V b = b u (w; r) + bb (e ) + ( bb ) V g : (B.9) In the steady state each location is equally attractive to workers and the expected value of migration e satis es the steady state version of the paper s equation (2): e = pv b + ( p) V g : (B.0) Note that equations (B.8), (B.9), and (B.0) form a linear system of three equations in three unknowns, V b ; V g ; and e. The solution to this system depends on the equilibrium prices w and r: The latter can be found by replacing the paper s rst-order conditions (0), (7), and (8) into the market-clearing condition for land, equation (9). The steady state version of the paper s equations (6) (8) yield the steady state gross ow rates: o = x = gg + p ( gg bb ) : (B.) As this equation shows, the steady state gross ow rates are related by simple expressions to the parameters that describe the stochastic process for the evolution of the idiosyncratic shock. I use this expression and the related one that describes the ve-year migration rate to calibrate the parameters gg and p. In the non-stochastic steady state of the model the measure of poorly matched workers in each location is m = px: In what follows I present the closed-form solution for the non-stochastic steady state of the model assuming that bb = and = ; as in the benchmark calibration of the model. Replacing n = and z = in equation (B.7) yields:

10 0 u = + A! : The steady state prices of labor and land are (equations B.6 and B.5): w = + A! ; r = A + ( ) : Solving the linear system of three equations (B.8), (B.9), and (B.0) in the three unknowns utilities V g ; V g ; and e yields: e = (p b ( gg ) + ( p) g ) u ( ( p ( )) gg ) ; ( ) ( p gg ) V g = gu + ( gg ) (e ) gg ; V b = b u + (e ) : For this triple to be part of an equilibrium in which well-matched workers do not migrate while badly matched workers do, parameters must be such that: V b < e < V g : It is straightforward to show that both inequalities are satis ed in the calibrated version of the model ( b = 0 and g = ). C Solution Algorithm and Details On Numerical Implementation In this appendix I provide detailed information on: ) The considerable computational advantage of assuming that b = 0 (section C.); 2) The solution algorithm used to solve

11 the model and the actual steps involved in calibrating the parameters (; u ) (section C2.). C. Discussion of the Importance of the Assumption b = 0 To appreciate the importance of this assumption, consider how one would solve for the law of motion S (s; ) (given the expected utility of migration, e) if b was strictly positive. A standard approach to solve this problem would involve guessing the law of motion of the state vector S (s; ) ; solving the Bellman equations (5) in the paper, and then updating S (s; ) using the mobility conditions () (2) and (3) (5) in the paper. The updating step is potentially quite lengthy because of the relatively large number of cases to consider at each point (s; ) of the four-dimensional state space. Speci cally, in ows x(s; ) might be positive or zero; the out ow probability of badly matched workers q b (s; ) might be zero, positive but less than one, or one; if q b (s; ) = and x(s; ) = 0; then the out ow probability of well-matched workers might be either zero or positive. The number of combinations to consider is seven. Further, there is no guarantee that the updating process S (s; ) would eventually converge. The assumption b = 0 simpli es the solution algorithm because it makes it possible to divide the latter in two simple steps. The rst step consists of solving for the equilibrium value function V g (s; ) : Since the value function of an agent with shock b is simply V b (s; ) = (e ) ; the equilibrium value function V g (s; ) can be found before solving for S (s; ) : To see this note that the expected future value of V g (s; ) in the Bellman equation (5) is bounded from below and from above. The lower bound is simply the value of out-migration, e : The upper bound obtains from the mobility condition (2) in the paper, as locations that attract workers must o er expected utility e to incoming workers. To compute it, simply replace V b (S (s; ) ; 0 ) = (e ) in equation (2) of the paper and solve the latter for the expected value function of a g worker:

12 2 V g (S (s; ) ; 0 ) Q(; d 0 ) = e ( p) + p : p The upper bound on the right-hand side of this equation gives the expected utility that a location has to o er to workers with good matches to be in the condition of attracting new workers. The knowledge of the upper and lower bounds for the expected utility of g workers implies that equation (5) in the paper can be solved independently of S (s; ). When these bounds are not binding, in fact, by de nition well-matched workers do not migrate (q g (s; ) = 0) and no new worker enters in the location (x(s; ) = 0). The second step of the solution algorithm consists of nding the equilibrium policy rules q g (s; ) and x(s; ) at points of the state space in which the upper and lower bounds of the expected value function of g workers are binding. This is done by solving equations () and (5) of the paper for x(s; ) and q g (s; ) ; respectively. The assumption b = 0 also simpli es this second step because, given that q b (s; ) = at all points in the state space, there are only three, rather than seven, cases to consider to solve for x(s; ) and q g (s; ): a location can either experience in ows or not and, if it doesn t, well-matched agents can either leave or not. Note that di erently from the case b > 0 no iteration on S (s; ) is involved at this stage. C2. Detailed Steps of Numerical Solution This part describes the steps that I follow in solving the model and calibrating the parameters (; u ). The algorithm is comprised of three loops: one for nding the value function of well-matched workers and the law of motion for the endogenous components of the state vector conditional on e and the parameter vector (; u ), one for nding the equilibrium value of e for given (; u ) ; and one for nding (; u ) in order to match the two empirical moments of interest. Every change in (; u ) entails new equilibrium values for e, while a new guess for e requires the computation of the associated value function.

13 3 Step (Guess). Start from an initial guess for the parameter vector (; u ) and for the value of search e: The guess for e is updated in Step 3 below, while the guess for (; u ) is updated in Step 4. Step 2 (Dynamic Programming). Solve the dynamic programming problem for g workers and the policy rules q g (s; ) and x(s; ). This is the most time-consuming step of the procedure because there are four continuous state variables in the problem (recall that s includes n and m while includes z and z ). This part of the algorithm generalizes the approach of Lucas and Prescott (974) to the case of two types of agents who are ex-post di erent. The assumption that b = 0 greatly simpli es the problem as the value functions of badly matched workers is simply V b (s; ) = (e ) in this case. There are two sub-steps: 2a) Solve for the equilibrium value function of g workers. 2b) Solve for laws of motion N (s; ) and M (s; ). Sub-Step 2a. Solve for the equilibrium value function of g workers. By de nition this value function solves: V g (s; ) = g u (s; ) + gg max V g (S (s; ) ; 0 ) Q(; d 0 ); e + ( gg ) (e ) ; (C.) where I have taken into account the closed-form solution for V b (s; ) : This functional equation can be further simpli ed by recognizing that when the expectation inside the max operator is larger than e (none of the g workers chooses to leave) two possibilities exist. Case. No new worker chooses to locate there. In this case S (s; ) is such that: N (s; ) = gg (n m) ; M (s; ) = 0 The expected value function of g workers must satisfy:

14 4 p (e ) Q(; d 0 ) + ( p) V g (S (s; ) ; 0 ) Q(; d 0 ) < e: (C.2) This inequality expresses the fact that no migrating worker will choose to migrate to this location. Notice that equation (C.2) can be rewritten as: V g (S (s; ) ; 0 ) Q(; d 0 ) < e ( p) + p : p Case 2. Some new workers choose to locate there. In this case equation (C.2) holds as an equality: V g (S (s; ) ; 0 ) Q(; d 0 ) = e ( p) + p : (C.3) p It follows from this discussion that the value function (C.) must satisfy the following functional equation: V g (s; ) = g u (s; ) + ( gg ) (e ) + gg min max V g (S (s; ) ; 0 ) Q(; d 0 ); e ; e ( p) + p ; p where the argument S (s; ) of the value function inside the max operator is known and given by: N (s; ) = gg (n m) ; M (s; ) = 0: Thus, iteration on this functional equation will give the equilibrium value function V g (s; ) : Sub-Step 2b. Given the two value functions one can solve for the law of motion of the state vector S (s; ) : Speci cally, x(s; ) must be such that an in-migrating worker is indi erent between locating in an island characterized by state (s; ) or elsewhere in the

15 5 economy: V g (S (s; ) ; 0 ) Q(; d 0 ) = e ( p) + p ; (C.4) p where S (s; ) is such that: N (s; ) = x(s; ) + gg (n m) ; M (s; ) = px(s; ): Thus, by solving equation (C.4) for the unknown x(s; ) we obtain the measure of workers migrating into a location with state (s; ): Similarly, q g (s; ) must be such that an outmigrating worker is indi erent between staying in an island and leaving: V g (S (s; ) ; 0 ) Q(; d 0 ) = e ; (C.5) where S (s; ) is such that: N (s; ) = gg (n m) ( q g (s; )) ; M (s; ) = 0: Thus, by solving equation (C.5) for the unknown q g (s; ) we obtain the measure of g workers migrating out a location with state (s; ): Points of the state space for which neither (C.4) or (C.5) hold do not either experience out-migration (by g workers) or in-migration: q g (s; ) = x(s; ) = 0: Step 3 (Equilibrium). Once the equilibrium law of motion of population N (s; ) in a location is known, it is possible to solve for the equilibrium value of e by imposing that the sum of workers across all islands equals one. In practice, de ne the function f (e): f (e) = n (ds; d) ; (C.6) where the integral term represents the total (average) measure of workers across islands.

16 6 Note that this is equivalent to the average measure of workers in a single island over an in nite number of periods since all islands are ex-ante identical. The integral in equation (C.6) is computed by simulating the economy for a very large number of periods (6 million) and computing the average population: T TX n t : t= The real number e such that f (e ) = 0 is computed using Brent s method for nding the zero of a function. Step 4 (Calibration). all the variables of interest. Given e ; it is possible to compute the equilibrium value of The vector (; u ) is estimated by constructing the model counterpart of the two moments listed in the text (Section 4) and choosing (; u ) so that the model-generated moments are exactly equal to their empirical counterparts. Since there are two parameters and two moments, this is an exactly identi ed model. The problem then becomes one of solving two non-linear equations in two unknowns. In order to nd a solution for this non-linear system of two equations in two unknowns I have used Broyden s algorithm. The latter operates in the following way (for a more detailed description, see Press et al. 996, chapter 9). First, it numerically approximates the Jacobian matrix associated with the non-linear system. It then uses this approximate Jacobian to nd an updated vector (; u ) by implementing the Newton step, which guarantees quadratic convergence if the initial guess is close to the solution. If the Newton step is not successful, the algorithm tries a smaller step by backtracking along the Newton dimension. When an acceptable step is determined, (; u ) is updated and the algorithm can proceed in the way described above, once an updated Jacobian has been obtained. Since the numerical computation of the Jacobian can be costly (and in this model it is), the Jacobian at the new vector (; u ) is iteratively approximated using Broyden s formula. The non-linear solver stops when the maximum percentage di erence between the simulated moments and the empirical moments

17 7 is, in absolute value, smaller than 0 4. References Card, D Immigrant in ows, native out ows, and the local market impacts of higher immigration. Journal of Labor Economics 9, Lucas, R., Prescott E., 974. Equilibrium search and unemployment. Journal of Economic Theory 7,