Human Capital Production in a Spatial Economy:
A Quantitative Assessment of the Decentralized U.S. Education System*

Sunham Kim†
January 29, 2024
Job Market Paper
Please click here for the latest draft.
Abstract
Education and labor mobility are key drivers in the production of humancapital, fundamental to economic development. In the U.S., the varying skill
production efficiencies of state-specific education systems and the dynamics
of worker migration shape human capital of states, influencing economic
outcomes at both the state and national levels. This paper develops a novel dynamic spatial general equilibrium model with overlapping generation framework in which heterogeneous individuals accumulate human capital andmove across states. Calibrated to the U.S. economy, the model illustrates
how variations in education efficiency lead to substantial cross-state income
disparities and shows that internal migration can notably boost output instates with lower education efficiencies. At the national level, free mobilityof workers yields a 6.9% output gain. Moreover, the model suggests that variations in human capital account for 46.6% of the state variation per capita
output. Applying the calibrated model to analyze the Obama Administration’s Race to the Top initiative finds that the $4.1 billion grant spurred a 0.2%
increase in U.S. GDP, mostly benefiting the grant-winning states and theirneighbors. Additionally, alternative grant allocation experiments show thatstrategic reallocation of education grants, considering state skill production
efficiencies, could further increase national GDP gains without necessarily
worsening state income disparities.
* I am deeply grateful to my advisors Chong Xiang, Mario Crucini, Farid Farrokhi, Anson Soderbery, Maria-Jose Carreras-Valle, and Soojin Kim for all their feedback and guidance. I thank Trevor
Gallen, Seungki Hong, Daniele Coen-Pirani, Nicholas Manuel, and Hyunjae Kang for helpfulcomments. I also thank seminar participants at Purdue Trade-Spatial Reading Group, Purdue
Macro-International Trade Brown Bag, MEA 2023, UEA 2023, Midwest Macroeconomics Meeting
2023, and SEA 2023. I thank the Purdue Research Foundation Research Grant and the Bilsland
Dissertation Fellowship for generous financial support. All errors are my own. Previously titled “A
Dynamic Model of Skill Formation and Worker Migration.”†
Kim: Purdue University, West Lafayette. Contact: sunham-kim@purdue.edu

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

1.

1

Introduction

Education and labor mobility jointly forge the production of human capital, thecornerstone of economic development. In the U.S., this interplay manifests
in a unique decentralized system of human capital production, where states
take the lead in setting policies and allocating financial resources. No state
stands alone; instead, each benefits from the influx of human capital produced
beyond its borders, provided that mobility is high across states. The literature
has examined elements of human capital production policies (Card and Krueger
1992; Chetty et al. 2014; Jackson et al. 2016; Bergman et al. 2019), migration
incentives (Dahl 2002; Nechyba 2006; Kennan and Walker 2011; Amior 2019),
and human capital production on a national scale, presuming a centralized
educational system within each country (Erosa et al. 2010; Manuelli and Seshadri2014). Yet the quantitative analysis of this interplay— namely, how one state’s
efficiency of human capital production propagates to the other states, and how
these spillovers contribute to the national economy—is a question that demands
a rigorous investigation.
To fill this gap, this paper develops a dynamic spatial general equilibrium
model with overlapping generations (OLG) of workers, which can be used toquantify the cross-state human capital flows and to evaluate state-level education
policies. In doing so, I make a couple of contributions. My main contribution is toshow mechanisms by which endogenous human capital production, interactingwith the migration choices of workers in general equilibrium, determines theflows and allocation of human capital across U.S. states, and thus affects thegross and per capita output of each state and the nation. Second, leveraging thecalibrated model, I conduct a cost-benefit analysis of a federal education grantthat subsidizes state education, by tracing out how the impacts of this policy
propagate across U.S. states. In this process, I show that the geographic allocation

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SUNHAM KIM

of the federal grant is pivotal in determining its effectiveness, with direct impacts
on human capital in the grant-winning states and general equilibrium effects on
the adjacent states and the national economy.
In the model, heterogeneous workers make forward-looking decisions onhuman capital acquisition and migration. Human capital acquisition is a costly
choice at the beginning of life. Each individual draws their innate ability atbirth and, observing this draw, chooses how much to invest in knowledge, withparents financing this investment. The state of birth’s skill production efficiency,
akin to the notion of ‘education quality’ in a broader sense, affects the amountof acquired knowledge; states with higher efficiency enable more knowledge
acquisition, given identical ability and investment. This innate ability combinedwith the acquired knowledge constitute an individual’s skill. Then, based on thisskill, each individual chooses whether to attend college, factoring in both thefixed and idiosyncratic costs of college in terms of utility. In sum, one’s humancapital is defined as a combination of skill and type of degree. Once the choice
made, workers move across states carrying fixed human capital. They supply
efficiency units of labor, which is equivalent to their skill, to a local labor market
defined as a degree-state pair. On the other hand, states differ in their efficiencies
in producing final good and producing skills.
Central to the model is the interplay between the two channels. When workers
make human capital decisions, they internalize the future possibility of migrating
to high-wage states.1 These anticipated migration choices, in turn, are affectedby each state’s efficiency in skill production, reflecting workers’ considerations
1

The expected income is a primary determinant of migration decisions. High-skilled workers
are more inclined to migrate and to make longer-distance moves. See, for example, Borjas et al.
(1992), Dahl (2002), Kennan and Walker (2011), Grogger and Hanson (2011),Young (2013), andAmior (2019).

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

3

for their expected offspring, conditional on their own human capital.2 This setup
produces rich individual policy functions. Migration propensity is higher amongcollege-educated workers compared to non-college workers, and within each
degree category, it increases with skill. Regarding skill investment, parents tend toinvest more if they have higher incomes or live in states with high skill production
efficiency, or if their child possesses higher innate abilities. In equilibrium, given
the individual policy functions, the markets for the final good and human capital
clear in each state at all times. While individual workers may migrate across states,
the overall net flows balance out in the steady state.
Solving this model is computationally intensive. States are heterogeneous.
Workers in each state differ by age, innate abilities, skill investment, types of degree, and states of birth. Each type of worker has different migration propensities
for each potential pair of origin and destination states in all periods. The model
must track all those state vectors and policy functions to characterize the spatial
equilibrium. To manage the computational demands, this analysis focuses on
the U.S. states and internal migration, although the framework can be applied
to a range of geographical entities.3 Additionally, I use a model-specific solution
algorithm to solve the model in a feasible time frame, exploiting the model structure where human capital is fixed once chosen. This algorithm facilitates thequantification of the model’s rich features while avoiding any simplification.
I calibrate the model to the 2000 U.S. economy. First, I estimate the bilateralmoving costs and migration elasticity by applying the two-step Poisson Pseudo
2

Parents seeking neighborhoods that offer superior educational opportunities, thereby investing in their children’s human capital, as documented by Bayer and Timmins (2007) and Nechyba(2006). This migration decision accounts for the enduring impact of neighborhood and educational quality on children’s economic outcomes, as highlighted in Chetty et al. (2016) and Jackson
et al. (2016), for example.
3
Also, educational spending varies significantly across states. Coen-Pirani (2015) reports thatover half of the differences in real K-12 education spending per pupil across U.S. school districts
stem from variations between states, rather than within individual states.

4

SUNHAM KIM

Maximum-Likelihood (PPML) estimation (Artuc 2013; Artuç and McLaren 2015)
to the average wages by state and the 5-year cross-state migration flows fromthe Current Population Survey. Second, I solve for the state fundamentals andparameters governing the human capital decision to clear all labor markets and tomatch data moments such as population share, wage, and per capita output. The
model not only matches the targeted moments perfectly, but also generates untargeted moments such as the national college population share close to the data
counterparts. Moreover, the average model-predicted migration propensities
across states are close to the observed ones.
I find substantial variations in model-inferred skill production efficiency
across U.S. states, with a ratio of 1.23 between the 90th and 10th percentiles,
indicating that the most efficient states produce 23% more efficiency units oflabor the least efficient ones, given the same skill investment. The skill production efficiency is positively correlated with public education spending per pupil,the moment not being targeted. Higher spending per-pupil is associated withhigher final good TFP, which in turn leads to an amplification of the variations inboth TFP and skill production efficiency through this association. In other words,
states efficient in skill production are likely to have high gross and per capita
output from both high human capital stock and high TFP. This is further amplified by endogenous skill formation since a portion of higher output translates
into skill investment, which again has positive feedback to output. Such positive
feedback loops in state economies may intersect significantly with cross-statehuman capital flows.
Migration plays a crucial role in reducing state income disparities, being
driven in part by preference shocks as commonly modeled in the literature. To
see this, I consider a case of Oklahoma, a state with both low skill production
efficiency and final good TFP. First, it imports workers embodied with higher

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

5human capital. Second, having a chance of migration induces higher humancapital investments for individuals in such states by raising expected returns,
thus increases state human capital stock and bolster output per capita by up
to 7%. Moreover, the reallocation of human capital due to migration effectively
boosts the overall efficiency of the U.S. economy. I find that the aggregate U.S.
output would decline by approximately 7%, equating to a loss of $700 billion in2000 dollars, absent worker mobility. This estimate stands as the first attempt toquantify the economic contributions of the state-level public goods in humancapital production, thereby illustrating the decentralized U.S. education system’s
benefit from high internal mobility.
The gains from human capital spillovers vary across states. For instance, thegains could be smaller for states more efficient in skill production such as New
York and Connecticut. While gross output of such states may increase as long as
net HC inflow is positive, their per capita may decrease if the in-migrants embody
lower human capital than the native residents. The calibrated model indicates up
to a 3.8% loss in per capita output for high skill production efficiency states, incontrast to the significant gains in less efficient states. These heterogeneous responses across states highlight the potential impacts of state educational policies
on gross and per capita output accounting for spatial linkages.
Using the calibrated model, I examine the impacts of a federal grant on state
education. Specifically, I evaluate Race to the Top (RTT) grant from the Obamaadministration on the regional and national economies. In 2010-2011, the Department of Education distributed $4.1 billion in competitive grants to states thatelected to participate, with the amounts awarded varying among the 18 states
and the District of Columbia that received funds. To gauge the program’s effects, I
simulate a scenario wherein the RTT grants are translated into enhancements inskill production efficiency.

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SUNHAM KIM

I find that the RTT initiative augmented U.S. output by 0.2%, which corresponds to an increase of $21.5 billion, marking the first rigorous cost-benefit
analysis of the program. The federal investment appears to yield sizable net benefits, $17.4 billion, partly due to the subsidy’s spillover effects to neighboring states
and the amplification of benefits. Specifically, output per capita in grant-receiving
states rose between 0.1% in Illinois and 0.92% in Tennessee, depending on theamount of subsidy per pupil. Furthermore, adjacent states enjoy output gains
as well, albeit smaller, ranging from 0.03% in Oregon to 0.14% in Vermont, because of higher human capital embodied by in-migrants from the grant-receiving
states. In sum, the ripple effect of the RTT grant benefits both the grant-receiving
states and their neighbors, with the magnitude of these benefits depending on
the geographical positioning of the recipients.
Finally, in exploring alternative RTT grant allocation strategies, I find that theallocation scheme can significantly influence economic outcomes at both stateand national levels. I first consider a hypothetical reallocation of the RTT funds
exclusively to the five states with the lowest education quality: North Dakota,
Oklahoma, New Mexico, Montana, and Mississippi. This allocation would resultin a 0.13% boost to U.S. output—less than the increase observed under the actual
RTT program’s implementation—but would also narrow the income disparities
among states. For instance, Oklahoma’s per capita output ratio compared to New
York, the wealthiest state, would rise from 0.56 to 0.65.
Conversely, directing the RTT grants to the top five states in terms of education
quality—New Jersey, Nevada, Washington, New York, and Connecticut—would
enhance U.S. output by 0.27%. This increase exceeds the gains from both theoriginal RTT implementation and the bottom states scenario. Remarkably, thisapproach does not exacerbate income inequality; the output ratio of Oklahoma
to New York increases slightly to 0.58. This can be attributed to the fact that

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

7subsidizing states with already high education quality could induce spillover
benefits through migration. With these states strategically positioned—three
on the East Coast and two on the West—the potential for spillover effects is
considerable, extending through neighboring states and infusing the broader
U.S. economy. This result illustrates that targeted investments in states withalready high education quality can yield national output gains without necessarily
widening income disparities across states. It also underscores the importance ofgeographical considerations in the allocation of federal education grants, as thespatial dynamics play a crucial role in amplifying the impact on the overall U.S.
economic output.
Related Literature This paper relates to several veins of literature. The mostrelated are papers that develop general equilibrium models with endogenous
human capital production, such as Erosa et al. (2010), Manuelli and Seshadri(2014), Cubas et al. (2016), Hsieh et al. (2019), and Xiang and Yeaple (2021). This
paper complements this class of models by accounting for location choices byheterogeneous agents who care about both economic and educational prospectsof locations. A key departure from the literature, among other dimensions, is thatagents internalize the possibility of future migration to other states in their humancapital decisions. While it shares commonalities with Ferriere et al. (2021), which
employs overlapping generation model to examine skill acquisition in responseto trade shocks, this study centers on the spatial allocation and cross-state flows
of human capital.
Recent studies have advanced dynamic quantitative spatial general equilibrium models in international trade and economic geography, addressing labormarket dynamics and endogenous migration. Notable works include Artuç etal. (2010), Desmet et al. (2018), Caliendo et al. (2019), Fogli and Guerrieri (2019),

8

SUNHAM KIM

Lyon and Waugh (2019), Giannone et al. (2023), and Crews (2023). This paper
complements these models by integrating worker heterogeneity and differencesin state skill production efficiencies, leading to endogenous skill acquisition dynamics. Specifically, building on Bryan and Morten (2019), and situated within an
extensive literature on selection, sorting, and migration (Roy 1951; Acemoglu andAutor 2011; Lagakos and Waugh 2013; Lindenlaub 2017; Hanson and Liu 2021),
this study probes the interactions between mobility and skill formation, theirconsequences for state and national output, and the implications for regional
education policy at both state and national levels.4
This paper connects to a large literature on the effects of education policies
and investments on economic outcomes of individuals and geographical entities,
such as Card and Krueger (1992), Fernández and Rogerson (1998, 1999, 2003),
Gordon (2004), Albouy (2012), Chetty et al. (2014), Coen-Pirani (2015), Jackson
et al. (2016), Altonji and Mansfield (2018), Bergman et al. (2019), Handel andHanushek (2023), and Biasi (2023). I contribute to this literature by providing
a general equilibrium framework, which is able to quantify the propagation of
education policies across geographic borders. By doing so, it can evaluate thespillover of state or local education policies as well as offer an angle of federalism.
For instance, as an application, I provide the first cost-benefit evaluation of Raceto the Top (RTT) initiative from the Obama administration. In contemporaneous
work, Eckert and Kleineberg (2021) propose a model in which residential and educational decisions result in geographic sorting, and abstract away from the roles
4

Hsiao (2021) explores similar themes using a static spatial framework and Indonesian data.
This paper focuses on the U.S. and presents a dynamic model of skill formation and migration
decisions. Bertoli and Rapoport (2015) and Delogu et al. (2018) are other papers in the similar
spirit that endogenize education and migration choices. While my paper exclusively focuseson internal migration in the U.S., both are in the context of international migration. A tworegion model presented by Bertoli and Rapoport (2015) focuses on the role of migration networks.
Delogu et al. (2018) study the impact of immigration restrictions such as visa policy on the world
distribution of income.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

9of innate ability and parental income in human capital choices, a key mechanismin this study.
This paper also relates to the literature that studies the U.S. spatial dispersion,
such as Berry and Glaeser (2005), Diamond (2016), Ganong and Shoag (2017),
Fajgelbaum and Gaubert (2020), and Rossi-Hansberg et al. (2019). While abstracting away from popular elements such as agglomeration, housing market, andfirms’ location choices, my paper contributes to this literature by elucidating how
human capital constitutes spatial dispersion through production and reallocation
processes.
Lastly, in the domain of development accounting that investigates the role ofhuman capital in income disparities, this paper is closely related to Hanushek
et al. (2017) who also focus on the U.S. states.5 They find variations in humancapital account for 20-30% of variation of per capita output across U.S. states,
based on new measures of human capital to account for cross-state migrations.
However, their work, based on data-driven measurement and accounting, differsfrom my research, which offers micro-founded measures of human capital stocks
and flows and delivers higher share of human capital, around 47%.

2.

Model

2.1

Environment

I consider an economy composed of multiple states indexed by k ∈ K. The
economy is populated by a continuum of overlapping generations of workers
who are altruistic towards their children. Each worker has one and only one
5
For the international application of development accounting focusing on human capital, see,for example, Mankiw et al. (1992), Klenow and Rodrı́guez-Clare (1997), Hendricks (2002), Caselli(2005), Hsieh and Klenow (2010), Schoellman (2012), Jones (2014), Schoellman (2016), Hendricks
and Schoellman (2018), and Rossi (2022).

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SUNHAM KIM

child throughout their life, thus each generation has the identical population size.
Workers possess different levels of human capital, which is characterized by skill
and degree. Skill is characterized by innate abilities ε and acquired knowledge
H, which is a function of state of birth kB , skill investment e. There are two typesof degree, college and non-college, indexed by o ∈ {c, n}. College education lets
worker access to high-paying jobs that requires a college diploma. It however
does not change one’s skill. Individuals have to pay fixed cost χ and idiosyncratic
cost z, in terms of utility. Regions differ in four dimensions: amenities ak , labordemand shifter Ako , the final good TFP Θk and the efficiency in skill production
hk . In each state, a representative firm produces non-tradable final good. Allfirms have a CES technology and demand labor from all types of degree. Time isdiscrete and indexed by t ∈ {0, 1, 2, · · · }.

2.2

Equilibrium

2.2.1

Households

There is a measure-one continuum of finitely-lived workers. Age is indexedby q = {1, 2, . . . , qR }. Workers are endowed with an innate ability ε, of which
distribution F (ε) has a strictly positive support. They work for qR periods andhave a child at age qP < qR . Fertility is exogenous and deterministic thus one adultgives a birth to one child. Workers derive utility from consumption of the final
good. Parents care about their children’s lifetime utility as well, hence financetheir skill investment.
Workers can move across states every period. They receive state-specific taste′

shocks uk at the end of each period and choose where to reside in next period.
The taste shocks are iid across time and workers. Relocation from k to k ′ incursodegree-specific moving cost τkk′ ≥ 0 measured in utility, where the equality holds

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

11

only if k = k ′ . Following Artuç et al. (2010) and Caliendo et al. (2019), I assume u
is a vector of K independent shocks that follow Type I Extreme value distribution
with zero mean and scale parameter ν. The taste shocks u and abilities draw ε are
independent.
Workers’ human capital is a combination of skill and degree. Skill is again
constituted by innate ability ε and acquired knowledge H. They draw innate
abilities at birth from LN (0, σε2 ). It is not genetically inherited, i.e., the ability of achild and her parents are not correlated.6 Parents Instead affect children’s humancapital through residential and skill investment decisions: parent’s state of residence is children’s state of education; parents finance children’s skill investmentby their income. They cannot borrow to finance for financing. A child with q = 0
is in the “education stage”, and invest in skill, and choose whether or not to go
to college, given the innate ability, state of birth, and parental transfer for skill
investment. Human capital is fixed over lifetime after the education stage.
The following skill production function, translates skill investment e to acquired knowledge for an individual born in state kB .7
H(kB ; e) = hkB eη .

(1)

In a broad sense, hk is a parameter that captures educational quality of statek. The parameter η ∈ (0, 1) is the elasticity of skill with respect to investment,
which would capture diminishing returns. In sum, one’s human capital stock is a
product of innate ability and acquired human capital, and it is the very amountof effective labor supplied by the worker. Workers earn a market wage for each6

The “genetic inheritance” assumption is popular in the papers that use the overlapping
generation setup to study intergenerational inequality, such as Erosa et al. (2010), Fogli and
Guerrieri (2019), Ferriere et al. (2021), etc. While I abstract away from this assumption, the model
is able to incorporate it.
7
The log-linear specification follows Hsieh et al. (2019), Xiang and Yeaple (2021), and Erosa etal. (2010).

12

SUNHAM KIM

supplied unit of effective labor, thus labor income is a product of the market wage
and effective labor:
I k (kB , ε, e, o) = wok Hε = wok (hkB eη )ε,(2)

where o is the degree type.
Households are forward-looking and have subjective discount rate β ∈ (0, 1).
The timeline for the household’s problem and decisions is as follows:
1. Children arrive and their abilities vector ε′ realizes. All households in state kknow the economic condition of the state;
2. All adults work and earn the market wage;
3. Observing ε′ , parents choose their own consumption and children’s skill
investment. Children choose degree type. Non-parent adults enjoy consumption;
4. The state-specific taste shock realizes and state choices are made. The
terminal age workers retire and exit from the labor market. Newborns enter
the labor market as a worker with age 1.
I formalize the household’s decision problem at each stage of their lifecycle. I
denote the state variables fixed over time by s ≡ (kB , ε, e, o) to ease the notation.
Let vq (k, s) be the lifetime utility of a worker of age q at time t in state k, bornin state kB with ability draw ε, invested e for knowledge, and chose degree o.
Also, let V q (k, s) ≡ Evq (k; s) be the expected lifetime utility where the expectation
is taken over the state-specific taste shocks. The utility function u(c) is CRRA,
1−ρ

u(c) = c 1−ρ−1 .8
8

The utility function choice has a nontrivial implication on workers’ decision rules on skill
investment and migration. See Appendix B.2.1.

13

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

Working Stage At the beginning of a period, adult workers observe the economic
condition of all states, work, and earn the market wage. They have the option to
relocate then. The worker’s problem formulated recursively as follows, where P k
is the final good price in state k.
vqt (k; s) = u(ckt (s)) + ak + max
k′ ∈K

n ′
o
q+1 ′
k
k,k′
ut − τo + βEvt+1 (k ; s)(3)subject to ckt (s) = Itk (s)/Ptk .
The first and second terms constitutes the flow utility. The third term is the′

future value of the market. It depends on the idiosyncratic taste shock uk , degree′

specific mobility cost τok,k , and the expected future value function for destinationlabor market k ′ . As usual in the discrete choice model literature (ACM, CDP, etc.),′

the Type I extreme value distribution assumption of uk lets me rewrite (3) as
follows:
Vtq (k; s) = u(ckt (s)) + ak + ν loghX ν1 i′q+1 ′exp βVt+1(k , s) − τok,k

(4)

k′

s.t.

ckt (s) = Itk (s)/Ptk .

I use the V notation for the rest of the paper unless v is necessary. It delivers amigration probability from k to k ′ for a worker with (k, s) as
h
i1
q+1 ′
k,k′ ν
exp βVt+1 (k ; s) − τomqt (k, k ′ ; s) =
h
i1 .
P
q+1 ′
k,k′ ν
k′ ∈K exp βVt+1 (k ; s) − τo(5)The probability depends on one’s human capital. I relegate the relationshipbetween human capital and m to Section 2.3.

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SUNHAM KIM

Education Stage (“pre-period”) At q = 0, children start in their parent’s stateand draw innate ability ε. Observing the draw, parents finance (thus choose)the level of skill investment e for children. Children then choose whether to go
to college, given (ε, e), subject to the fixed cost of college χ and idiosyncratic
cost of college z. The state-specific taste shock realizes at the end of the period,
then newborns choose the initial state to join as a worker with q = 1. Thus,
the newborn’s problem consists of two sub-problems. Let Vt0+ (kP , ε, e, o) be theexpected lifecycle utility of a child who draw ability ε, spend e for human capital,born in kp (p for parents), and choose degree type o. Formally, the end-of-period
problem is written as follows9 :Vt0+ (kP , ε, e, o) = ν loghX ν1 i′1
exp βVt+1(k ′ , s) − τokp ,k

(6)

k′

That is, children face the same state choice problem as the workers, but have noown consumption.10 In the beginning of the education stage, the value of a childin kp with (ε, e) who chooses her degree type o ∈ {c, n} is given as
Vt0− (kP , ε, e) = maxo∈{c,n}no
Vt0+ (kP , ε, e, c) − (χ + z), Vt0+ (kP , ε, e, n) .

(7)

The child pays a cost of college χ + z in terms of utility. The first term χ is a fixed
cost common to everyone. The second term z is a random cost, which distribution follows Logistic(0, σz2 ). The optimal degree-choice policy o is obtained9

As in the working stage, before solving for the expectation, the problem is written as
h
n
oi
k
kP ,k
1
v0+
(k
,
ε,
e,
o)
=
E
max
u
−
τ
+
βv
(k;
s)
P
t
t+1
k

10

This assumption is common in the literature, e.g. Erosa et al. (2010), Fogli and Guerrieri (2019),
Eckert and Kleineberg (2021). The parental consumption could be interpreted as a household
consumption.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

15

from solving (7) and the initial labor market choice m0 is obtained from (6). It
determines the measure µq=1(k ′ ; s) of type s workers in state k ′ at time t.
t
The last building blocks of the household’s problem are the decision problems
for retiring workers (q = qR ), parents (q = qP ), and the workers who will become
parents (q = qP − 1). Those are “special” periods in the working stage.
Terminal Period

In the terminal period, a worker work, earn, consume, andretire:VtqR (k; s) = u(Itk (s)/Ptk )(8)Equation (4) is a simple a hand-to-mouth problem. The retiring workers work,
earn, consume, then exit the labor market. They have no relocation choice.
Parental Period

The parent’s problem connects the children and adults. In

the beginning of the parental period, children arrive with their innate ability ε′ .
Observing ε′ , parents face a tradeoff between their own consumption c and child’s
skill investment e′ . Formally, they solve the following problem:
VtqP (k, ε′ ; s) = maxu(ckt (s)) + ak + ν log
′
c,e

hX


 ν1 i′qP +1 ′
exp βVt+1(k , s) − τok,kk′+ αVt0− (ε′ , e′ , k)
s.t.

(9)

ckt (s) + (e′t )k (s) = Itk (s)/Ptk .

Compared to Equation (4), Equation (9) has the fourth term, which is a valuefunction of child. The parameter α governs the degree of alturism. Notice that the
new ability draw ε′ enters into the state space of the parents as they fully observe

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SUNHAM KIM

its level. On the other hand, In the meantime, the workers who will become
parents make migration decision before knowing ε′ , as will soon be discussed.
Pre-parental Period Pre-parental adults are the workers who will become aparent in the following period. (q = qP − 1). They do know they will be a parent,
observe child’s ability pair ε′ , and make the skill investment decision in the following period. But they do not know the level of ε′ thus take expectation over ε′ incontrast to parents. As a result, their future value of the market is different from
Equation (4), the problem of non-parent adults whose age is neither qP nor qR .
VtqP −1 (k; s) = u(c) + ν loghX ν1 i′exp βEε′ VtqP (k ′ , ε′ , s) − τok,k

(10)

k′

ckt (s) = Itk (s)/Ptk .

s.t.

While the ability of (future) children is integrated out, VtqP (k ′ ; s) reflects the′difference of hk across states. Conditioning on s, a worker is likely to relocate′

to a state with higher hk which delivers higher EVtqP via higher future value ofthe child V00− , all others equal. In sum, the relocation decision account for both
worker’s own job prospect and child’s expected lifetime utility, which is a functionof human capital. The migration probability can be derived likewise to (5).
The worker’s problem defined above jointly generates a set of policy functions
as follows. For 1 ≤ q ≤ qR , cqt (k; s) denote consumption; For 0 ≤ q ≤ qR − 1:
mqt (k, k ′ ; s) denotes migration probability from k to k ′ ; For q = qP , (e′ )qt P (ε′ , k; s)denotes skill investment for children; For q = 0, o(ε′ , e, kp ) denotes degree choice.
2.2.2

Production

Production is a standard static representative firm’s problem. In each state, arepresentative firm hires workers in both college and non-college in order toHUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

17

maximize output with the following CES technology. The time index t is omitted
to ease the notation.
σ
h
i σ−1
k
kD σ−1kkD σ−1σσ
Y = Θ Ac (Lc )+ An (Ln )
,

k

k

(11)

where Θk is state k’s output TFP, σ is the elasticity of substitution, and Aki (i = c, n)
and the CES demand shifters (Akc + Akn = 1). The degree-specific effective labors
are not perfectly substitutable, but workers are perfectly substitutable within each
degree cell. Lki (i = c, n) is the effective labor demanded by the final good producer.
The final good is non-tradeable. The labor demand schedule is standard.
k σ−1
LkD
o = (Θ ) wk −σ
o
Yk
k
k
Ao P

(12)

Given the market wages of each degree (wck , wnk ), the competitive producer’s cost
minimization yields the price of the final good:
Pk =
2.2.3

i 1
1 h k σ1−σk σ1−σ 1−σ
(A
)
(w
)
+
(A
)
(w
)
c
n
c
n
Θk

(13)

Equilibrium

I close the model with the following market clearing conditions.
Measure Define µqt (k; s) to be the measure of type s agents age q in time t. Let Se
and Sε be the space of skill investment and innate ability, respectively. Define thespace space S = Se × Sε and B the σ− field on S. Each age group has the uniformR
PR P
qsize q1R . The measure of workers is normalized: qq=1
kB ,k,o B dµt (k; s) = 1 forany t. The measure of children in state k before drawing abilities and making any
P Rchoice is identical to the measure of parents: µ0t (k) = kB B dµqt P (k; s).

18

SUNHAM KIM

Local Goods market clearing (spot) The total expenditure on the final goodand equals the total consumption and skill investment.
Ytk = Ctk + Etk ,

(14)

where
Ctk =

X Z
[ cq (·)dµqt ],q≥1,kB

Etk =

B

XZ
kB

Z

Sε′

e′t dµqt P

B

Note that E is integrated over the measure of parents, because they choose e′ .
Local Labor Market Clearing (spot)

This should be straightforward: no excess

demand in the labor market.
kD
LkS
ot = Lot∀o,k,t

(15)

where
LkS
ot =

X

LkS
ot (q)

(16)

q≥1

LkS
ot (q) =XhZ
kBe )εdµqt

kB η

(h

i

(17)

B

Notice that the summation in (17) is with respect to state of birth kB given thestate of residence k.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

Law of motion: global evolution of the labor force

19

We need an equation thatsummarizes the aggregate labor market evolution.
′µq+1
t+1 (k , s) =X

mqt (k, k ′ ; s)µqt (k, s)∀t,

0 ≤ q ≤ qR − 1

(18)

k

That is, for q ≥ 1, labor supply in state k ′ at time t + 1 is a sum of effective labor ofstaying workers and in-migrants from all states.
Equilibrium At every period, endogenous outcomes are determined by the
distributions of labor (so human capital). The distribution is determined by the
dynamic problem of workers and the constant fundamentals of the economy,
which I denote by Γ. It consists of the total factor productivity Θk , the skill pro′ductivities hk , amenities ak , CES demand shifters Ako , and moving costs τok,k . The
structural parameters in the model are given by the risk aversion ρ, the migration
elasticity ν, the dispersion of the innate ability distribution σε , the elasticity ofskill production η, college fixed cost χ, college idiosyncratic cost dispersion σz ,and the CES substitution parameter σ. I define a stationary equilibrium as follows.
With the paper’s analysis centered on steady states, time subscripts are omitted.
Definition 1. Given the fundamentals Γ, a stationary equilibrium of the model
consists of prices {wok , P k }, value functions and policy functions for workers{V q (k; s)}, c(k; s), e′ (k; s), o(k, e, ε), mq (k; s), policies for firms {LkDo }, and a stationary measure µq (k; s) that solves the dynamic problem of individuals (4),(6), (7), (8), (9), (10), and the corresponding laws of motion for labor (18),sub-static subproblem for production (11), (12) (13), and market clearing
conditions (14), (15).
In English, a stationary equilibrium describes a situation in which no aggregate
variables change over time. Individuals may move from one market to another,

20

SUNHAM KIM

but the net flows are zero. Appendix D.2 is a complete description of my solution
procedure for the equilibrium.

2.3

Qualitative Predictions

This subsection describes the qualitative predictions of the model compare it withthe existing literature. In order to unpack the intuition transparently, I consider asimplified partial equilibrium model. I specifically consider workers live throughthree periods (qR = 3), after one period of education as a child. I shut downcollege choice to keep the discussion concise.
2.3.1

Skill Investment in the OLG setup

The OLG structure in the model follows a common practice of the literature in thata working parent makes an investment decision. The implementation is somewhat non-standard in that the continuation value of children is considered in the
middle of the parents’ lifecycle. Therefore, it is useful to clarify skill investment
incentives and under which conditions those are operative.
I focus on the skill investment choice e′ . There is no migration for now, thus
the worker’s state vector s ≡ (e, ε). V q (e, ε) is a value of a worker with age q,
skill investment e, and ability ε. Wage w is exogenous and the final good priceis normalized to 1. The worker’s problem then can be written as the following
finite-horizon dynamic programming problem.
V 3 (e, ε) = u(I)V 2 (ε′ ; e, ε) = maxu(I − e′ ) + αV 0 (e′ , ε′ ) + βV 3 (e, ε) s.t. c + e′ = w(heη ε) (19)′c,eV 1 (e, ε) = u(I) + Eε′ βV 2 (ε′ ; e, ε)V 0 (e, ε) = βV 1 (e, ε).

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

21

Differentiating (19) with respect to e′ , we have the FOC that governs the investment choice, which is the heart of skill formation:
uc (I(e, ε) − e′ ) = αVe0′ (e′ , ε′ )(20)

The LHS represents the parental side of the choice by the marginal cost of skill
investment due to the foregone consumption of parents. The RHS, the child’sside of the choice, the marginal value of the skill investment, is fundamentally afunction of the discounted sum of the marginal utility of consumption of children
throughout their lifetime. In other words, (20) is the Euler equation that governs
the consumption-skill investment decision.11
Parental income, which is a function of (e, ε), and the child’s ability ε′ are
two main factors that determine the optimal e′ . A higher parental income, froma higher parental H, shifts it up (“parental income effect”). A higher ability ε′
implies a higher HC given e′ , and so does higher consumption, thus shifting it
up (“child ability effect”). Those recast standard lessons of an intertemporalconsumption choice. Consider the child’s ability effect first. A higher ε′ implies ahigher income for the child and a rise in the child’s consumption in all periods.
(income effect) It changes the relative prices of the parent’s own consumption
in the current period and the child’s lifetime consumption. (substitution effect)
In other words, the child ability effect is a version of the (intertemporal) priceeffect. The parental income effect, on the other hand, changes a degree of theintertemporal substitution effect by changing the marginal utility of parental
consumption, which eventually changes the relative price of consumption andinvestment, given ε′ . Since both sides are a function of marginal utility, the curvaThis equation cannot be rearranged for a closed-form solution of e′ as the RHS involves V 0 .
Still, it can be shown that the FOC has a unique solution under many different setups as long as
value function is concave.
11

22

SUNHAM KIM

ture of u(c) has substantial implications on the e′ choice. In a nutshell, under the
CRRA setup, the model needs ρ ∈ (0, 1) to generate reasonable qualitative predictions. It makes the substitution effect dominates the income effect. Otherwise,
the RHS decreases in ε′ , making a weird prediction that parents spend less for
smarter children.12
Comparison against reference papers

The HC production function closely

follows Hsieh et al. (2019). Human capital investment is increasing in innate
ability. That is, the HC production function is not supposed to be sensitive toutility function choice to this extent. The sensitivity to the utility form comesfrom the OLG setup. In Hsieh et al. (2019), the key problem is to maximize the net
income after skill investment. The HC production function governs the marginal
benefit of investment, and the marginal cost is nothing but the unit price ofinvestment. Utility functions thus play no salient roles there. In my model, bothmarginal benefit and cost are calculated through the utility function given the
OLG structure.
2.3.2

Skill Investment in Spatial Economy

I now introduce geography in the model. There are two states, A and B, andworkers can move between those. Migration is driven by idiosyncratic taste shockand moving cost τ . Wages are exogenous, but differ across states. I consider
wA < wB without loss of generality. In order to focus on the impact of geographyon human capital choices, I let skill production efficiency h be identical between
the states.
In words, allowing migration lets workers have a chance to earn from statesother than they are born and educated. It changes the expected returns to human12

See Appendix A.1 for derivations and risk aversion choice.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

23

capital compared to the no migration case. The option to access more lucrative
markets encourages low-wage state parents to spend more on their children’s skill.
It is the “American Dream” part of the migration. On the other hand, migration isa downgrade risk to the high-wage state parents. It thus discourages them andthe optimal human capital choice becomes lower than the case where children
stay in the high-state forever. A formal argument starts by rewriting the worker’s
problem and deriving a new Euler equation. k is a state index.
V 3 (k, e, ε) = u(I k )V 2 (k, ε′ ; e, ε) = maxu(I k − e′ ) + αV 0 (k, e′ , ε′ )
′
c,e
hX

1/ν i+ ν log
exp βV 3 (k ′ , e, ε) − τ Dk′ ̸=k

(21)

k′

c + e′ = wk (heη ε)hX1/ν iexp βEε′ V 2 (k ′ , ε′ ; e, ε) − τ Dk′ ̸=kV 1 (k, e, ε) = u(I k ) + ν log
s.t.

k′

V 0 (k, e, ε) = ν loghXexp βV 1 (k ′ ; e, ε) − τ Dk′ ̸=k1/ν i(22)k′Differentiating Eq. (21), we obtain a new Euler equation that echoes Eq. (20):uc (I k (e, ε) − e′ ) = αVe0′ (k; e′ , ε′ )(23)Equation (23) clarifies that geography works only through children’s value V 0
since the future value of parents does not affect the investment decision. The
LHS, the marginal cost, is the same as the no-migration Euler equation. The RHS,
the marginal benefit, is now a function of all possible wage flows throughout thechild’s life. To make this point clear, I differentiate Eq. (22) with respect to e, for a

24

SUNHAM KIM

child in state A:

1/ν

1/ν
exp βV 1 (A, e, ε)βVe1 (A, e, ε) + exp βV 1 (B, e, ε) − τ
βVe1 (B, e, ε)0
Ve (A, e, ε) =

1/ν
P
1
′
k′ exp βV (k ; e, ε) − τ Dk′ ̸=k= m1 (A, A)βVe1 (A, e, ε) + m1 (A, B)βVe2 (B, e, ε)

(24)

′

where mk,k is a probability of migration from k to k ′ . In the world of no freedomof movement, mAB = 0 and Ve1 (A, e, ε) is a function of human capital and wA .
In the migration world, mAB > 0, and now the marginal value of investment
becomes a function of both wA and wB . Since the marginal value of investment
is higher in B, thanks to wA < wB , A-born children now enjoy a higher expectedmarginal value of investment. (“American Dream”) It shifts up the RHS of theEuler eqn thus inducing higher investment, ceteris paribus. The story goes in
the opposite direction to B-born children. Their expected marginal value ofinvestment decreases, compare to the no migration case.
In other words, migration “averages” the marginal value of investment acrossstate and make state optimal choices converge to some extent, but not completely.
This result also has an important implication for the TFP-induced amplification
mechanism. While amplification is still operative in the migration world, its degree would be weaker compared to the no-migration world, because the increaseof skill investment is smaller due to the discouragement. It is also notable thatthe drop in the high-wage state and rise in the low-wage state can be asymmetric,
depending on utility form: the rise is greater than the drop under quadratic utility.
It is related to the HC gradient of migration propensity, which will be discussed in
Section 3. In short, the rise in the low-wage state is large if the gradient is positiveand steep. Going back to Equation (24), the two components of the second term

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

25

of the RHS are increasing in Hε, so skill investment is the one stone that kills two
birds.
Discussion: Does migration discourage skill investment in high-wage states?
It may sound unrealistic that migration discourages skill investment in high-wage
states. It could be seen as a result of taste-shock-driven migration. However, (i) if
migration happens in every period, (ii) a probability of migration from a high- to
low-wage state is non-zero, it is a simple consequence of the tradeoff between
parental consumption and children’s human capital. Consider a parent in B, thehigh-wage state. She has to decide how much to spend on her child. She knowsthat the child has to leave B tomorrow in 100% probability. Then, it is optimal
to choose as if she is a parent in A and spend more on her own consumption. In
other words, it is about a choice facing a probability of getting into an undesired
state.
The “discouragement” happens even if migration is purely selection-based.
Suppose that there is an exogenous threshold of skill for B-educated children tostay in B 100%. In this case, again, parents with low-ability children will choose asif they are in A because they know that the child has to leave tomorrow. It means
more heterogeneity in degree choice across abilities given the state of birth andshows there always exists discouraged parents unless one can permanently live inthe high-wage state. Thus, the discouragement is a feature that stems the natureof migration, not from the current modeling strategy.
2.3.3

Migration Propensity and Human CapitalIn this subsection, I describe individual migration patterns by one’s human capital, such as selection and sorting. It also includes a tradeoff between wage andeducation quality that all workers to be parents face. I start with the two-state

26

SUNHAM KIM

partial equilibrium setup in the previous section first and consider an extra state
if needed. Here, all selection and sorting through skill occur within a given degree
type. A difference in migration propensity across degree types is generated bydegree-specific moving costs.
Selection I first study selection. I define there is a positive selection if high-skill
workers are more likely to move to high-wage states. It is useful to consider age
qR − 1 worker s who is in moves from A to B and will retire tomorrow. The skill′

q=2(k,k ;s)is as follow:
gradient of migration propensity ∂m ∂(Hε)∂mqR −1 (A, B; s)β= mqR (A, A; s)mqR (A, B; s)∂(Hε)ν
h ∂V qR (B; s) ∂V qR (A; s) i
−
×
∂(Hε)
∂(Hε)
|
{z
}

net gain of migration from marginal value of HC

It is intuitive that the worker is more likely to move to B if the net gain is positive.
′

Since V qR (k ′ ; s) = u(ck ) where ck = wk Hε, I can rewrite the net gain term as
follows:∂V qR (B; s) ∂V qR (A; s)−= woB u′ (cB ) − woA u′ (cA )
∂(Hε)
∂(Hε)

(25)

which shows that the marginal value of skill is a product of its unit price, wage,
and marginal utility of consumption. It is clear that utility form specification
again plays a pivotal role here. Given wA < wB , the direction of the net gain isgoverned by the degree of diminishing marginal utility of consumption. The net
gain is positive only if the wage gain from migration is greater than the decrementof marginal utility.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

27

I have shown that high-skill workers are more likely to move into the highwage state, namely B. The flip side of the coin is, they are less like to move into the
low-wage state, namely A. It is a natural question to ask whether they are more
likely to move compared, in general. It can be analyzed using the skill gradient of
the probability of leaving current state k, 1 − m(k, k). In short, compared to lowskill ones, high-skill workers are less likely to leave k if k offers a high marginalvalue of skill but more likely to leave if there are better outside options. It is
straightforward to calculate the gradient:∂[1 − mqR (k, k)]= −mqR (k, k)(1 − mqR (k, k))
|
{z
}
∂(Hε)
(−)

×

h ∂V qR (k; s)∂(Hε)|∂V qR (k; s) im (k, k )−(26)
1 − mqR (k, k) k′ ̸=k
∂(Hε)
{z
}


1

X

qR

′

(∗)

Note that the last term of the RHS is the expected marginal value of skill investP
ment because 1 − mqR (k, k) = k′ ̸=k m(k, k ′ ). Thus, (∗) is the difference between
the marginal value of investment in the current state and the expected marginalvalue of skill in other states, which I called the “outside option.” If the current
state offers a higher marginal value of skill, (∗) is positive, and the whole RHS
is negative, meaning that the probability of leaving k is decreasing in skill. The
direction goes opposite if the outside option is better than staying.
In sum, higher-skill workers are more responsive to marginal value differential:
they are more likely to move to seek higher marginal value, which is coming fromhigher wages. It is how the model features selection.
Sorting Now I consider sorting across destinations. The same intuition applies.
Consider a third state C and the first derivative of the odds ratio between mi-

28

SUNHAM KIM

gration propensities from A to B and A to C, mqR (A, B; s) and mqR (A, C; s), withrespect to skill:
h mqR (A, B; s) i β h ∂V qR (B; s) ∂V qR (C; s) i
∂
log
−
=
∂(Hε)
mqR (A, C; s)
ν
∂(Hε)
∂(Hε)

(27)

Here, the direction is again determined by the relative marginal values of humancapital. If wC < wB , the bracket term in the RHS is positive, and high-skill workers
are more likely to sort into B, compared to C.
Tradeoff for Parents: Wage versus Education Quality I have been considering
the case of workers who will retire tomorrow. Workers who are going to havea child tomorrow face another tradeoff: high wage does not necessarily comewith high education quality. If a state has high wages and high education quality,
they reinforce each other and increase the migration propensity to the state.
But if a state has high wages and low skill production efficiency, or vice versa, it
depends on the relative sizes. It would be best illustrated by the total derivative of
migration propensity from k to k ′ of workers to be parents tomorrow.
dmqP −1 (k, k ′ ) =∂mqP −1 (·) k′ ∂mqP −1 (·) k′
dw +
dh
∂wk′
∂hk′

(28)

where both partial derivatives in the RHS are positive. (See Appendix A.1 forderivation.) Equation (28) is a mathematical representation of the intuition that′′

I described above. If dwk > 0 and dhk > 0, the total differential is definitely
positive. But once their signs differ, the whole direction depends on the size. It isa distinctive incentive of migration for parents.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

2.3.4

29

General Equilibrium Mechanisms

I explain the general equilibrium mechanisms in the model and relate those to
the individual-level behaviors stated in previous sections.
Characterizing Equilibrium

The equilibrium is characterized by state funda-

mentals (Θk , hk ) and moving costs. The moving cost is a key friction to feature
persistent geographic wage differentials in the steady state. (Cameron, Chaudhuri, and McLauren 2007) Diminishing returns to scale is working as a congestion
force here that prevents every worker moving into the highest-wage state in thepresence of persistent wage differentials. It also implies that a state must have ahigh TFP in order to host a high HC stock, by paying them high wages.
Amplification 1: TFP-induced A higher Θ translates into a higher w, inducing ahigher e′ that further increases the gross output Y and per capita output y = Y /N ,all others equal. This positive feedback is the first amplification mechanism themodel features. In the presence of migration, a higher w induces a higher HC
inflow, which is likely to be a result of the inflow of high-skill workers, from theselection at the individual level. It further boosts the gross output and gross skill
investment. However, increasing state HC stock (from both amplification andmigration) compresses the state wage, which while gross output and investment
are increasing, the directions of per capita output and investment are ambiguous.
Amplification 2: HC-induced

Without migration, h works as a part of TFP

in the sense that the aggregate labor supply L is a linear function of state h.
Thus, a higher h induces higher investment, which yields a higher gross output
thus higher investment. The positive feedback is weaker than the TFP-induced
amplification since the marginal product of labor declines and there is no initial

30

SUNHAM KIM

boost in state wage, compared to the TFP-induced amplification. In other words,
the total factor productivity is a dominant force because high h-educated workers
are likely to move into high Θ states that offer higher wages. Moreover, skill
production efficiency h only affects migratino decision of workers who will be aparent tomorrow.
Amplification 3: Migration-induced Net inflows of human capital is anothersource of positive feedback cycle of gross state output growth from higher HC
stock, additional skill investment, and additional output growth. However, thedirection is ambiguous in per capita term, because the inflow lowers the marginal
production of human capital. In terms of migration, a higher h induces a higher
inflow of parents, which also pushes down the state wage. So the same intuition′

applies. It is useful to consider Equation (28) in this perspective: dhk > 0 de′crease dwk , considering the GE effect, thereby quantitatively limiting the state’s
advantage of having superior education quality.
Amplification 4: Heterogeneity-induced

College and non-college workers are

an imperfect substitute. Therefore, a higher stock of college-level HC boosts
non-college wages, induces in-migration and generates another cycle of positive
feedback.

2.4

Leveraging Model-inferred Regional Human Capital

2.4.1

Adjusted Net Inflow: Gains from Human Capital Migration

The model quantifies state human capital stock and flows. I leverage the modelinferred quantities to study the state human capital gains (losses) from migration and the contribution of in-migrants to state economy. I introduce several
additional notations: population by current residence N k , output per capita

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

31

y k ≡ Y k /N k , population and human capital by state of birth L̃k and Ñ k , respectively. Moreover, human capital carried by workers from s, who are now in d isdenoted by Ld (s). I omit the degree subscript o for now to ease the notation.
Suppose that there are two states A and B. Rewriting the local labor market
clearing condition, we have A’s human capital stock as a sum of A-born, inmigrants, and out-migrants, which is analogous to the GNP, import, and exportin the national accounting identity:
Current HC stock

A-born workers

z}|{
LA

z }| {eL(A)=in-migrants

out-migrantsz }| {z }| {+ LA (B) − LB (A)(29)

The model-inferred L quantifies the net human capital inflow LA (B) − LB (A). It
is different from the net population inflow N A (B) − N B (A). Region A may sufferfrom brain drain under the zero net flow of population (N A (B) = N B (A)) if theout-migrants possess higher human capital (LA (B) < LB (A)). In general, for agiven observed migration flow measured in headcounts, there can be infinitely
many possible combinations of embodied human capital in state-born, inflow,
and outflow. A high HC stock of state k could be from either its own high HC
efficiency or net positive inflow. The population flow, which is observable, is nota perfect measure to deterimne the human capital flows, while it helps the model
pin down the per capita HC by source and destination.
Thus, I propose the gross and per capita HC accounts in spatial economy as
follows.
e
Lk = L(k)

+

X
s̸=k

Lk (s)−X

Ld (k)

(30)

d̸=k

k

ek h Lek X N k (s) Lk (s) X N d (k) Ld (k) i
L
N
=
+
−
ek
e k N k (s)e k N d (k)
Nk
Nk N
N
N
s̸=k

d̸=k

(31)

32

SUNHAM KIM

Equation (30) is a multi-state version of Equation (29). It is about the total volume
of human capital flows. It is related to the state output. Equation (31), adjusts thevolume using the size of k−born population.
The model plays a crucial role in characterizing the structural HC flows, whichneeds both N and L. The former is observable from the real-world data. L has tobe measured through the model, especially to be consistent with the endogenous
components. It is the second and third terms in the bracket in (31). I close
this subsection by defining the net human capital inflow in terms of per capita
measure, Adjusted Net Inflow (ANI) for future use. We now have the degree
subscript o back.
AN Iok ≡X N k (s) Lk (s)
o

s̸=k

=

eok
N

X N d (k) Ld (k)ooo−eok Nod (k)Nok (s) d̸=k N(32)

1
N GIokke
No

The Human Capital Flow and Regional Education

The model-inferred L showsa state’s working population does not reflect its h. Recall that migration shuffles
workers across spaces, and so does hk . Let’s denote a j-educated HC stock in k by
Lk (j) = hj L̂k (j). L̂ is a function of investment and ability, which would be used byan econometrician to measure the state’s human capital stock.13 It lets me write
Lk =X

Lk (j) =jX

hj L̂jjh X  L̂j  i X
j
L̂j
=
P j h
j L̂
j
j
|
{z
}
≡ĥk

13

Consider an econometrician who predicts state HC stock using individual expenditure, controlling unobserved heterogeneity (“ability”) through a framework such as a dynamic factor
model.

33

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

where ĥk is the “migration-averaged” HC efficiency in that it is an weighted
mean of hj by total efficiency units by source j. The second summation termof the last equality is a function of skill investment and ability, which are thevariables employed by the econometrician. Clearly, hk ̸= ĥk . For example, if hk is
the highest (lowest) among states, ĥk underestimates (overestimates) the state’s
education quality. It shows that ĥk cannot be taken as an education quality ofstate k. My structural framework lets us to back out hk consistently, going beyond
data-driven accounting exercises which accounts for population flows.
2.4.2

Measuring In-Migrants’ Contributions in the Economy

My model also quantify the in-migrant’s contribution in gross output. It is seemingly straightforward: under perfect competition and perfect substitutability
across origin given degree, income shares deliver the contribution of Lko (kB ) in
the neoclassical sense.
Yk =

X
o

wok



Lko (k)| {z }

k-born k-residents+ Lko (−k)| {z }

(33)

all migrants

where −k denotes any source states other than k, i.e. Lko (−k) ≡Pks̸=k Lo (s).

The income share of all migrants with degree o is then calculated by ϖok (−k) ≡wok Lko (−k)/Y k . It however does not identify the value-added contribution of
migrants. Suppose that all economies are symmetric and Lko (s), the migrants’human capital, is identical across s. As a result, no state economies obtain
additional output boosts from migration, no matter how high (or low) ϖok (−k) is.
Any changes in the migrant income shares ϖok (−k) are mechanical.

34

SUNHAM KIM

It is now clear that gross and value-added contribution of migrants should
be distinguished as in goods and services trade.14 The notion of value added
contribution again calls for the per capita representation. After some algebra,
Equation (33) can be rewritten in per capita terms, as a weighted sum acrossdegree and origin:
Emp shares by degree
k

y =

X
o

wok

z}|{
Nok
Nk

Nok (−k) Lko (−k) io+Nok (k)Nok Nok (−k)

h N k (k) Lk (k)
o

Nok
|

{z

}

k-born k-residents

where

|

{z

all migrants

(34)

}

X N k (s) Lk (s)Nok (−k) Lko (−k)
o
o
≡
−k
k
k
No
No (−k)No Nok (s)s̸=kThe per capita income is decomposed into the per capita HC by degree and origin,weighted by degree employment shares and within-degree origin composition. I
define the Relative Human Capital Per Capita (RHCPC) of migrants in k as a percapita HC ratio of migrants and natives:
RHCP Cok =Lko (−k)/Nok (−k)−1
Lko (k)/Nok (k)(35)

Simply put, RHCP Cok is the excess per capita human capital of migrants. It is positive (negative) if the migrants possess a higher (lower) per capita HC comparedto the native residents. RHCP Cok = 0 if the migrants and natives have the same
per capita HC. The underlying thought experiment is: which level of per capita
output would k have attained if the economywide per capita HC were the same
14

The notion I have in mind is the double-counting problem and measurement of value-added
content of goods trade.(Hummels et al. 2001; Johnson and Noguera 2012) Here, I frame the “purely
shuffled” HC analogous to the “double counted” domestic content of exports.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

35

as the native residents, given w? Denoting by ȳ k the counterfactual per capita
output, the comparison is between y and ȳ k .
It is worthwhile to clarify a conceptual difference of AN I and RHCP C. Clearly,
AN I is an absolute measure related to gross output Y , while RHCP C is a relative
measure related to per capita output y. The more important difference is theunderlying logic. The former compares the in-migrants against the out-migrants
while the latter does against the stayers. Suppose that the latter follows the former. The underlying thought experiment compares the current per capita output
against a counterfactual per capita output before HC reallocation, i.e. y and ỹ.15
As ỹ is calculated as if there is no labor reallocation between economies, thisalternative comparison is about the additional gain from featuring endogenous
migration on top of endogenous human capital accumulation. It is a relevant
question for quantitative evaluation of each model component (comparison
between equilibrium outcomes), but not for measurement of value added contribution of migrants given equilibrium.

2.5

Development Accounting

It is straightforward to derive the following accounting equation from Equation
(11):
1=

Cov[ln y, ln g(Lc , Ln )] Cov[ln y, ln Θ)]+V ar[ln y]V ar[ln y](36)where g is the CES aggregator without the TFP . Equation (36) is a standard development accounting equation, which decomposes cross-state per capita income
variations into the TFP and the factor variation. Using the model-inferred L
15
Technically, the counterfactual wage w̃ would be different from w and ỹ should be adjusted
accordingly. I deliberately ignore it as the key is the choice of counterfactual here.

36

SUNHAM KIM

seamlessly accounts for the amplification effects via endogenous skilll formationand migration in the decomposition.

3.

Taking the Model to the Data

In this section, I describe how I take the model to the data. I begin by briefly
describing the data sources. I then detail how I identify the model’s structuralparameters with a combination of estimation and calibration strategies. Finally, I
show the quantification results of the U.S. states

3.1

Data

This subsection summarizes the data sources and measurements used to take themodel to the data. All further data descriptions are in Appendix C.1. All monetary
values are expressed in 2015 U.S. Dollars and normalized with respect to the 2000
median wage, $42,119.
Region Choice, Regional Output, and Working Population

I start with 47 U.S.

states. I exclude Alaska, Delaware, Wyoming, and the District of Columbia torule out potential confounding following Hanushek et al. (2017). I use the 2000
GDP by State data of the Bureau of Economic Analysis (BEA). As the model has no
capital, I trim capital quantitatively, assuming the Cobb-Douglas with labor andcapital and the labor’s share θ = 0.604 (Hsieh et al. 2019). Specifically, suppose
Y = Lθ K (1−θ) . I use Ŷ = (Y /K 1−θ )1/θ for calibration. I take the state-level capital
data from El-Shagi and Yamarik (2019). I use total employment of the BEA data
for the state population, given the model assumes all adults are employed.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

Wages, Migration Flows, College Shares

37

I construct 5-year migration flows

across states for each degree type using the 1985-2010 Current Population Survey
Annual Social and Economic Supplement (CPS-ASEC) complied by Flood et al.
(2023) and 1980-2000 Decennial Census (Ruggles et al. 2023). See Appendix C.2
for details. I calculate the average wages by degree and state from the same data.
I use the wage and migration flows to estimate moving costs and the migration
elasticity. For calibration, I use the 2000 data. The national population share ofcollege graduated worker is taken from the 1990 U.S. Census.

3.2

Model Parameters

This subsection discusses the estimation and calibration strategies of the model
parameters.
3.2.1

Estimated Parameters′

The migration elasticity 1/ν and the moving cost τok,k are the key parameters thatgovern the migration layer of the model. I use the CPS-ASEC (Current Population
Survey-Annual Social and Economic Supplement) data to estimate the parameters. I estimate it by using a two-step Poisson Pseudo-Maximum Likelihood(PPML) regression, following Artuc (2013) and Artuç and McLaren (2015).16 The
first step exploits the variations in bilateral gross migration flows to identify thecontinuation values and the moving costs. The first step regression equation isderived from Equation (5):h
τ k,l iZtq (k, l; s) = exp destqt (l; s) + origtq (k; s) − o + ζtq (k, l; s)ν16

See Appendix D.1 for all the details and derivations.

(37)

38

SUNHAM KIM

Table 1: Model Parameters
Parameter

Description

A. Estimated Parameters
1/νMigration elasticity′
τok,kMoving costs

Value

Source

1.62

PPML estimation, Table 2
PPML estimation, Table 3

B. Internally Calibrated Parametersσε
Innate ability dispersion
χCollege Attendance Fixed Cost
σz
Idiosyncratic College Cost Dispersion
ak
Amenityhk
Human Capital Production Efficiency
ΘkTotal Factor Productivity
CES demand shifter
Ako

0.05
0.916
Calibration
0.737
Calibration-0.004 (mean), Calibration, Table E.2
1.866(mean), Calibration, Table E.2
1.758 (mean), Calibration, Table E.2
0.700 (mean), Calibration, Table E.2

C. Externally Set Parametersα
AltruismβSubjective Discount Factor
ρ
Risk AversionσElasticity of SubstitutionηElasticity of Human Capital Investment

0.5
0.9
0.9
1.5
0.103

Del Boca et al. (2019)Equiv. to 0.98 (annual)
Chetty (2006)
Cantore et al. (2017)
Hsieh et al. (2019)

Mean moving costs are by degree (non-college, college).

where Ztq (k, l; s) is the headcount of individuals moving from k to l, destqt (l; s) is
a destination fixed effect, origtq (k; s) is an origin fixed effect, τ k,l /ν is the moving
cost normalized by the migration elasticity, and ζtq (k, l; s) is an error term. The
second step uses the first stage results and the wage variations across states anddegrees to identify the migration elasticity, estimating the following equation
derived from (4):
 I k (s) t+1+ ζtq (k, s)
k
ν
Pt+1

β
κqt (k, s) = Dtq (s) + u(38)

where κqt (k, s) is a function of Ntq (k, s) and the fixed effect estimates, Dtq (k, s) istime dummy and ζtq (k, s) is the error term. As in the previous studies, the crosssectional migration flows convey information on expected future values that are

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

39

a function of future wages and option values. Moreover, future migration flows
are sufficient statistics for the option values of the state labor market.
Setting β = 0.9, I obtain the preferred 1/ν estimate 0.617, which implies
ν = 1.62. (Table 2) It is slightly lower but comparable to the annual estimates
CDP’s 2.02 for the U.S. commuting zones and Caliendo et al. (2021)’s 2.3 for′European countries. The mean of the preferred τ k,k estimates are 7.54 and 6.56
for non-college and college, respectively. (Table 3)
Table 2: The Migration Elasticity
(1)
All

(2)
Non College(3)
College

1/ν

0.617***(0.142)0.546***(0.188)0.643***(0.141)

N

1880

940

940

The difference of college and non-college
estimates are statistically insignificant.

Table 3: The Estimated Moving Costs

Non-College
College

Mean

SD

Min

Max

7.234
6.450

1.744
1.304

3.081
3.307

12.13910.712See Appendix C.2 for the sample definitions.
Non-college and College stand for “High Schoolor less” and “Some College or More”, respectively.

40
3.2.2

SUNHAM KIM

Calibrated Parameters

I calibrate the fundamentals {hk , Θk , Ako , ak } (188 parameters) and the college
choice parameters {χ, υ} (2 parameters), given the estimated migration-relatedparameters. My data targets are state per capita GDP y k , population N k , average wage by degree and state wok , and the national population share of college
graduates. All others equal, (y, w) identifies the TFP Θk via the CES aggregatorand human capital demands. Higher (y k , wok ) implies higher Θk . Amenities ak ,which clearly makes states more attractive, adjust to target N k . The human capital
production efficiency hk governs the educational investment and thus humancapital supply. It adjusts to clear all state labor markets, where per capita labor demand Lko /N k is characterized given the data moments and Θk . The CES demand
shifter Ako adjusts to match the relative labor demand Lkc /Lkn to the relative supply.
A higher per capita human capital carried by college workers implies a higher
Akc /Akn . Lastly, χ and υ jointly match the national share of college graduates. A
higher χ makes college education expensive, thus decreasing the college share
but increasing the per capita HC of college workers, given wage premia wck /wnk .
The dispersion of idiosyncratic cost governs the college enrollment behavior byskill. A higher dispersion implies that any high (low) human capital child goes
(does not go) to college, thus equalizing the per capita HC of college and noncollege workers. Thus, χ and υ jointly pin down the college population share at
the national level and the implied per capita HC. Lastly, the standard deviationof labor income pins down the innate ability dispersion σε . Appendices D.2 andprovide the full detail of the calibration and model solution method.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

3.3

41

Quantification: U.S. States

I calibrate the model to the 2000 U.S. economy. I include 47 U.S. states, excluding
Alaska, Delaware, Wyoming, and the District of Columbia, following Hanushek etal. (2017).
3.3.1

Model Fit

The model exactly matches the state-level moments: per capita GDP, wages bydegree, and population shares. (Appendix Table E.1) It also closely matches thenational population share of college graduates. Moreover, while not being directly
targeted, migration rates by degree are close to the data counterpart. (Table 4)
Table 4: Model Fit: College and Migration Rates

College-graduate population share
Migration Rate (Non-College)
Migration Rate (College)

3.3.2

Model

Data

0.668
0.085
0.135

0.614
0.071
0.122

Model-Inferred TFP and Human Capital Production Efficiency

The model-inferred TFP and h are highly correlated. (Figure 1)
Validating the Model-Inferred Parameters The main goal of the calibration
is to back out the human capital production efficiency, hk and the final good
TFP, Θk . I first examine the validity of model-inferred h using public education
expenditure per pupil. education-related variables as a proxy of education quality.
I consider a simple OLS regression of h on public education expenditure per

42

SUNHAM KIM

Figure 1: Model-Inferred TFP and HC Production Efficiency

pupil:
hk = b0 + b1 Expenditurek + εk ,where Expenditure can be either level (in terms of 1,000 USD) and log variables.
Using the 1990 and 2000 expenditure data, Table 5 reports that h is positively correlated with expenditure per pupil. A $1,000 USD increase in per pupil expenditureleads to 0.031-0.038 unit increases in h. In terms of percentage, a 1% increasein per pupil expenditure leads to 0.387-0.378 unit increase in h. Combining thisto the interpretation of h, a unit converted to efficiency units of labor when $1
spent on human capital investment of households, it means a 1% increase inpublic expenditure per pupil is associated with nearly 40% higher human capital
supplied in the state.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

43

Table 5: h and Public Education Expenditure per Pupil
Level

Log

(1)
1990

(2)
2000

(3)
1990

(4)
2000

Expenditure Per Pupil

0.031***(0.006)

0.038***(0.010)0.387***(0.072)0.378***(0.101)R-squared

0.329

0.224

0.331

0.214

Robust standard errors in parentheses. ∗p < 0.1, ∗ ∗ p < 0.05, ∗ ∗ ∗p <0.01.

Table 6: Model Validation: TFP

Rank Corr.

1980

1990

2000

2014

0.280
(0.060)0.703(0.000)0.743(0.000)0.580(0.000)In addition, the model-inferred TFP is positively correlated with the existing
estimates from the literature. There are positive rank correlations between thecalibrated Θ and the 1980-2014 estimates taken Herkenhoff et al. (2018) . It isnotable that the stronger correlations between the model and the 1990 and 2000
estimates, 0.703 and 0.743 respectively, given that the model is calibrated to the2000 U.S. economy.
3.3.3

Model-Inferred Human Capital Flows

I now use the human capital flows quantified via the lens of the model to disentangle the migration flows in terms of headcount and the implied human capital.
Figure 2 visualizes the AN I by states and degrees. Similarly, Figure 3 visualizes
the RHCP C by states and degrees.1717

For levels, See Figures E.1-E.2 in the appendix.

44

SUNHAM KIM

Figure 2: Human Capital Flows: ANI by Education(a) High School or Less(b) Some College or More

Figure 3: Human Capital Flows: RHCPC by Education(a) High School or Less(b) Some College or More

45

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

3.3.4

Development Accounting

Finally, I use the model-inferred state human capital stock Lk to quantify theshare of per capita GDP variations explained by human capital. Table 7 reports
a decomposition result from the accounting equation (36). The human capitalvariations across states explains 46.6% of per capita GDP variations. It is morethan twice as large as the literature, such as 22.8% of Hanushek et al. (2017). The
difference comes from the way to measure state human capital stock. Hanushek etal. (2017) is an accounting exercise with a data-driven measure of human capital.
They build an index based on observed educational outcomes such as years ofschooling and test scores. They account for migration by weight-averaging the
education outcomes of state of birth for each state’s working population. My result
is inferred from the general equilibrium framework that systematically accounting
for migration and its impact on skill formation. Second, Hanushek et al. (2017)
has no amplification mechanisms. My model incorporates multiple amplification
channels of skill formation, skill heterogeneity, and migration discussed in Section
2.3.4. Those channels can generate large variations of per capita output from tiny
variations of TFP and skill production efficiency.
Table 7: Development Accounting: U.S. States

explained share of per capita GDP variations

4.

TFP

Human Capital

0.534

0.466

Counterfactual Exercise

In this section, I implement counterfactual analysis in several dimensions. First,
I shut down migration to quantify its role in the state and national economies.

46

SUNHAM KIM

Second, I consider a placed-based education reform. I boost the human capital
production efficiency h of a state. It could be interpreted as increasing public
educational spending on K-12.

4.1

A House Divided: No Migration Economy′

To implement the no migration exercise, I assign τok,k = 3, 000 and solve formarket-clearing wok vector that clears all labor markets, given the other state
fundamentals {Θ, A, a}. The resulting human capital distribution across states
let me calculate the counterfactual per capita GDP of all state economies and the
U.S. economy.
The national output drops by 6.9%. It is amplified through the channels that
had boosted the national economy. For example, since skill formation is endogenous, lower income leads to lower skill investment, which generates lower labor
supply, and so on. On the other hand, the state-level responses are heterogeneous. Figure 4 shows the changes in per capita output from free mobility to
no migration economy. The states with low h, such as Oklahoma, North Dakota,
Montana, etc suffers from substantial losses up to 7%. Clearly, those have been
benefiting from human capital carried by in-migrants. The states with high hbenefits from isolation, as they fully utilize human capital produced within theborder. Figures 5 and 6 illustrate the negative correlation between the changes
in per capita output and AN I and RHCP C by degree in baseline economy. It isnotable that net population inflow shows no or weak correlation.

4.2

Evaluation of Race to the Top Grant Program

Race to the Top (RTT) is a $4.1 billion competitive grant program of the U.S.
Department of Education, implemented by the Obama administration. The grant

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

47

Figure 4: Changes in per capita Output

amounts vary across states from $18 millions (Colorado) to $700 millions (New
York). I use the calibrated model to evaluate the direct and general equilibrium
impact of the grant on the state and national economies. Moreover, I examine if
an alternative grant allocation scheme could outperform the original.
4.2.1

RTT in the Calibrated Economy

Simulation Procedure I first convert the RTT grants to per pupil expenditure.
It is interpreted as an exogenous subsidy. I take the regression results in Table 5,
0.0381, to further translate the per pupil values to changes in h ≡ ∆h. Finally, I
add ∆h to the basline h to obtain hsim .
Table 8 is a summary of RTT grant-winning states, funding amounts, and thecorresponding ∆h.
Simulated RTT The U.S. output gain is 0.2%. Figure 7 illustrates the state-levelchanges. Output per capita increases in every states. The grant-winning states
enjoy higher output per capita, depending on the size of the per-pupil grant. The

48

SUNHAM KIM

Figure 5: Lost Human Capital Flows and Changes in per capita Output (nonCollege)(a) ANI(b) RHCPC(c) Net Population Inflow

Figure 6: Lost Human Capital Flows and Changes in per capita Output (College)(a) ANI(b) RHCPC(c) Net Population Inflow

spillover effect is concentrated to the adjacent states. As a result, the easternstates are more likely to enjoy output gains.
4.2.2

Alternative Grant Allocation

In this subsection, I examine the RTT grant allocation in terms of national output
gain. The grant clearly boosts the recipient states’ per capita output. It does not

49

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

Table 8: Race to the Top
Grant ($ Mill.)$ per pupil% to annual spending

∆h

25
18
700
400
75
43
17
17
250
250
38
700
400
400
41
75
500
100
75

28.5
24.8
287.5
276.8
406.8
21.0
25.5
22.9
293.1
256.4
28.9
242.9
309.2
218.0
22.6
476.7
550.0
872.0
1088.1

0.38
0.28
3.46
2.97
4.58
0.20
0.31
0.28
2.63
2.00
0.19
1.68
3.62
2.14
0.20
3.80
7.18
7.23
6.71

0.001
0.001
0.011
0.011
0.015
0.001
0.001
0.001
0.011
0.010
0.001
0.009
0.012
0.008
0.001
0.018
0.021
0.033
0.041

Arizona
Colorado
FloridaGeorgia
Hawaii
Illinois
KentuckyLouisiana
Maryland
Massachusetts
New Jersey
New York
North Carolina
Ohio
Pennsylvania
Rhode Island
Tennessee
DelawareDistrict of Columbia

My quantification sample excludes Delaware and the D.C.

imply that the RTT allocation maximizes the national output gain. I experimentwith two alternative allocation schemes. First, I allocate the grants to the fivestates at the bottom of h distribution. Allocating the total RTT budget to thosestates increases their per pupil spending by $2,315.6, thus ∆h = 0.088. Second,
I allocate the grant to the top five h states. It increases their spending by $647.0
and ∆h = 0.025. Table 9 summarizes the alternative allocation schemes.
Table 9: Alternative Allocation Schemes
StatesBottom 5
Top 5

ND, OK, NM, MT, MS
NJ, NV, WA, NY, CT$ Per Pupil∆h2315.6
647.0

0.088
0.025

50

SUNHAM KIM

Figure 7: Race to the Top and Output per Capita

Subsidizing Bottom 5 Figure 8 shows the output per capita change with respect
to the baseline. The grant again benefits the five grant-winning states and the adjacent ones. The five states’ output gains vary from 3.25% to 3.8%. The per capita
output ratio between New York, the richest state, and Oklahoma is 0.58, which isslightly higher than 0.56 in the baseline. That implies that federal education grant
can be a policy device to reduce income equality across states. It comes with a
0.13% U.S. GDP gain.
Subsidizing Top 5 Figure ?? illustrates the output per capita change after subsidizing Top 5, compared to the baseline. The five states’ output gains are from
0.8% to 1.1%, which is lower than the Bottom 5 case. The resulting NY-OK output
ratio, yOK /yN Y stays the same to 0.56, thus boosting the top 5 does not aggravatesthe state income inequality. The U.S. GDP, however, increases by 0.27%, which is
as twice larger than the Bottom 5 scheme. There are two main factors driving thedifference. First, the Top 5 states have superior Θ, thus stronger amplification andspillover, compared to the Bottom 5. Second, compared to the RTT and Bottom 5

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

51

Figure 8: Changes in output per capita: Subsidizing the Bottom 5

case, there are now two separate “clusters” in the East and West coasts. It implies
more states benefit from spillover, and it goes into the higher national per capita
output gain. In sum, this result illustrates the range of possibility of federalism in
education, in the world where workers move around carrying human capital.

5.

Conclusion

This paper proposes a unified framework of quantifying human capital stockand flows across spaces. Calibrating the model to the U.S. states, I find that skill
production efficiency varies across states up to 20%, which results in cross-state
income disparities. Migration-driven reallocation particularly benefits states withlower skill production efficiencies, such as North Dakota and Oklahoma, boosting
their per capita output up to 7%. Overall, the national output gain from the freemobility of human capital is 6.9%. Moreover, the model suggests that variations
in human capital account for 46.6% of the state variation per capita output. I use
the calibrated model to analyze the impact of Obama Administration’s Race to52

SUNHAM KIM

the Top grant. The $4.1 billion program yields a 0.2% higher U.S. GDP, mostly
benefiting the grant-winning and adjacent states. Alternative grant allocation
schemes could yield varied output gains at both regional and national levels.
While these findings underscore the significance of education systems in state
economic development, allocating grants to states with lower skill production
efficiency may not yield the maximum output gain at the national level.
My paper, however, does not argue that states are the primary locus of variation. Aside from the computational challenges, it is imperative to probe thedynamics at more granular levels, like cities or commuting zones. Nor does it
dismiss the relevance of international migration, which might offer a more precise
quantification of a state’s human capital stock by accounting for global humancapital flows. Nonetheless, this conclusion strongly implies that considering the
spatial dimension is essential for a comprehensive understanding of the production of human capital, especially in large, decentralized education systems likethat of the United States.
Moving forward, this work paves the way for a series of intriguing questions
to investigate further. This finding suggests a need for further investigation
into the long-term implications of declining U.S. internal migration on future
economic growth through human capital production (Molloy et al. 2011, 2014;
Kaplan and Schulhofer-Wohl 2017). Along the same lines, understanding how
to tailor federal grant allocations to amplify economic growth invites a closer
examination of the interplay between federal education policies and regional
economic development.
Extending the model to other large economies, such as the European Union,offers a fertile ground for further study. Here, the nuances of migration policies
and educational investments within the EU—akin to international dynamics
due to the union’s distinct structure—pose fascinating scenarios for analysis.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

53

Examining the Erasmus Programme, for instance, could yield insights into how
education subsidies influence labor mobility and economic integration among
EU member states. Future research in these areas holds not only academic value
but is also essential for policymakers navigating the landscape of regional andeconomic policies within large, integrated economies.

54

SUNHAM KIM

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Appendix
A.

TheoryA.1

Derivation of Model Predictions

This section builds on the simplified setup in Section 2.3. For the following paragraphs, I would abuse the notation to ease the notation to make the point crystal′∂u ∂cclear. Notice that Ve0′ (·) is a discounted sum of weighted sum of ∂c′ ∂e′ , where

the weight is the migration propensities. Thus it is a linear combination of thederivative. I replace the RHS with the derivative, ignoring the linear combination′∂c′ η−1 ′
structure. Moreover, from c = I = wHε, ∂e
ε , the marginal increase′ = ηwh(e )in consumption due to additional investment, does not depend on the utility
form.
1 1−ρ
Log, Linear, CRRA Utility Suppose u(c) = 1−ρ
c . The FOC becomes(wheη ε − e′ )−ρ = αh ∂u ∂c′ i∂c′ ∂e′= α × (wh(e′ )η ε′ )−ρ ×

∂c′
∂e′

= η(whε′ )1−ρ (e′ )η(1−ρ)−1
Both sides are a function of ρ.
1. Log utility: let ρ = 1. We have1wheη ϵ − e′

=

αη
e′

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

63

The LHS is a decreasing function of (e, ε), the determinant of parental HC.
However, the RHS is no longer a function of children’s ability ε′ because of
the combination of log utility and log-linear HC production function. It
mutes the ability effects. It is also notable that the RHS is not even a functionof wages, which would have a substantial implication on investment andmigration choices in the spatial economy.
2. Linear utility: let ρ = 0. In this case, the story goes to the opposite of the log
case. The FOC becomes
1 = αηwh(e′ )η−1 ε′
The RHS is an increasing function of ε′ , so the ability effect is working. On
the other hand, The income effect is not operative as the LHS is no longer afunction of parental HC.
3. CRRA utility: It features both effects. The LHS is similar to the case of log
utility, by definition of CRRA and log utilities, and the parental income effectis just fine. However, the ability effect may go in the wrong direction. If
ρ > 1, a higher ε′ decreases the marginal value of educational investment,
which implies a reverse ability effect. It requires ρ ∈ (0, 1) to be in the right
direction.
Quadratic Utility In short, both effects are operative here and in the right direction. Suppose u(c) = − 21 (c − c̄)2 where c̄ is a sufficiently high number. The FOC
becomes′ηc̄ − e − whe ε = αh ∂u ∂c′ i∂c′ ∂e′= α(c̄ − wh(e′ )η ε′ ) × (ηwh(e′ )η−1 ε′ )64

SUNHAM KIM

The LHS is decreasing in (e, ε) (income effect). The RHS is increasing in ε′ (ability
effect). Why? Compared to the linear utility, the LHS has a curvature since thequadratic utility is twice differentiable. Compared to the log utility, the RHS
remains to be a function of ε′ as the utility function does not convert the loglinear HC into an additively separable object. It is straightforward to show that
the RHS is increasing in ε′ as long as c̄ is sufficiently high.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

Figure A.1: Value and Marginal Value of Child by utility formVe(e, )

Value Function(a) Log utility

Educ Inv

Educ Inv
Low

High

Ve(e, )

Value Function(b) Linear utility

Educ Inv

Educ Inv
Low

High

Ve(e, )

Value Function(c) CRRA utility, ρ = 1.1

Educ Inv

Educ Inv
Low

High

V e (e, )Value Function(d) CRRA utility, ρ = 0.9

Educ Inv

Educ Inv
Low

High

V e (e, )Value Function(e) Quadratic utility

Educ Inv

Educ Inv

65

66

SUNHAM KIM

Figure A.2: The Euler Equations by utility form(a) Log utility (No changes in MB)

MC, MB

MC, MB(b) Linear utility (No changes in MC)

Investment
MC, Low Income Parents
MC, High Income Parents

Investment
MB, Low Ability ChildMB, High Ability Child

MC, Low Income Parents
MC, High Income Parents

MC, MB(d) CRRA utility, ρ = 0.9

MC, MB(c) CRRA utility, ρ = 1.1

MB, Low Ability ChildMB, High Ability Child

Investment

Investment
MB, Low Ability ChildMB, High Ability Child

MC, Low Income Parents
MC, High Income Parents(e) Quadratic utility

MC, MB

MC, Low Income Parents
MC, High Income Parents

Investment
MC, Low Income Parents
MC, High Income Parents

MB, Low Ability ChildMB, High Ability Child

MB, Low Ability ChildMB, High Ability ChildHUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

B.

Utility Forms and Migration Patterns

B.1

Net Marginal Gain of Migration

67

Recall that wA < wB . Since the terminal-age worker has no future value, it isstraightforward that
∂V qR (B; s) ∂V qR (A; s)−= wB u′ (wB Hε) − wA u′ (wA Hε)
∂(Hε)
∂(Hε)

(39)

Under the four utility forms, the RHS can be calculated as follows.
1. Log utility: there is no gain. Again, any wage change in k ′ serves as a priceeffect. It is muted under the log utility.
wB
wA−
wB Hε wA Hε
1
1
=
−
Hε Hε

wB u′ (wB Hε) − wA u′ (wA Hε) ==0

2. Linear utility: net gain comes from moving to a high-wage region. Its downside is, clearly, the gain is not a function of HC level (e, ε).
wB u′ (wB Hε) − wA u′ (wA Hε) = wB − wA
>0

68

SUNHAM KIM

3. CRRA utility: net gain comes from moving to a high-wage region only if
rho ∈ (0, 1). The gain is a positive but decreasing function of H.
wB u′ (wB Hε) − wA u′ (wA Hε) =

wB
wA
−
B
A
hw Hε w Hε
B 1−ρ

A 1−ρ

= (w )− (w )

> 0 if ρ ∈ (0, 1)

< 0

i

(Hε)−ρ

if ρ > 1

4. Quadratic utlity:wB
wA
w u (w Hε) − w u (w Hε) = B− A
hw Hε w Hε
B ′

B

A ′

A

i
= (wB )1−ρ − (wA )1−ρ (Hε)−ρ
> 0 if ρ ∈ (0, 1)

< 0

B.2

if ρ > 1

More Gradients

The expectation is taken with respect to the child’s ability that will be realized
tomorrow.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

69

Wage gradient of migration propensity
qP∂mqP −1 (·)
qP −1
′ ∂EV
=
m
(k,
k
)
′
∂wk
∂wk′
h 
k′ 
∂V qP +1 (k ′ ; ·) i
′ ∂c
= mqP −1 (k, k ′ ) E u′ (ck ) k′ + mqP (k ′ , k ′ )
∂w
∂wk′
|
{z
}

(40)

(+)

>0
where the expectation is taken across ε′ . The bracket term is positive. The firstterm is trivially positive. The second term is positive because, given qR = qP + 1,
∂V qP +1 (k ′ ; ·)′= u′ (ck ) × heη ε′k∂wwhich is positive. This results holds for other age q as long as V is an increasing
function of w, which shall be true under any reasonable environments.
h gradient of migration propensity
qP∂mqP −1 (·)
qP −1
′ ∂EV
=
m
(k,
k
)
∂hk′
∂hk′
h ∂V 0 (k ′ ; e′ , ε′ ) i= mqP −1 (k, k ′ ) E
∂hk′

(41)

>0
where the bracket term is the expected marginal value h for children, which isalways positive.

70
B.2.1

SUNHAM KIM

The Risk Aversion Choice

Recall the FOC with respect to the investment choice.
uc (I(·) − e′ ) = αVe0′ (·)
As both sides are a function of marginal utility, the curvature of u(c) has
substantial implications on the model predictions of optimal investment choice.
Table B.1 summarizes all results from common utility forms, given the log-linear
human capital production function. Figure B.1 visualizes the child ability (firstcolumn) and parental income (second and third columns) effects. See Appendix
A for the algebra and additional plots.
Table B.1: Utility Forms and Underlying Forces

Logarithmic
LinearQuadratic
CRRA

Parental income effectChild ability effect

Y
N
Y
Y

N
Y
Y
Y only if risk aversion ∈ (0, 1)

The parental income effect is operative as long as the marginal utility of consumption is decreasing. Thus it is working for all but linear, where the marginalutility is always 1. The child ability effect is governed by both HC production
and utility function. Recall that HC production is linear in ϵ′ .18 Thus, the effect is
operative in the desired way as long as the decreasing rate of the marginal utilityof consumption is lower than 1. For log utility, it is (ε′ )−1 , so exactly cancels out.
For linear, the marginal utility of consumption is 1 so the marginal increase of
HC production, which is again linear in ε′ solely governs the child ability effect.
18

This is linear. The HC technology is log-linear in e, and linear in h and ε.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY71

For quadratic utility, the substitution effect dominates so the RHS is increasingin ε. Specifically, the marginal utility of consumption and the marginal increase
in HC are linear in ε′ so the RHS is increasing quadratically. Lastly, for CRRA,
the RHS becomes a function of ε1−ρ where ρ is risk aversion. That is, if ρ ∈ (0, 1),
substitution effect dominates income effect but otherwise. So, if ρ > 1, RHS isdecreasing in ε′ , making a weird prediction that parents spend less for smarterchildren.

C.

Data

C.1

Data Sources

to be added

C.2

Construction of Migration Flows

Previous studies used two sources of migration flows: (i) Census or equivalent
datasets19 (ii) individual panel datasets.20 The census-type surveys are moresuitable for my goal. Those are designed to cover the entire U.S. and have large
sample sizes, which is helpful to construct a migration matrix. However, they do
not keep track of the same individuals and the datasets have to be reshaped as a
pseudo-panel. The panel datasets’ strengths and weaknesses are the opposite.
The two-step PPML estimation needs the flow data from at least two consecutive periods, preferably three or more for lagged instruments. For this reason,
using the census-type surveys is not straightforward. One approach with the 519

Borjas et al. (1992); Dahl (2002); ACM; Molloy et al. (2014); Kaplan and Schulhofer-Wohl(2017); Amior (2019); CDP; Eckert and Kleineberg (2021), etc.
20
Kennan and Walker (2011) use the NLSY and Kaplan and Schulhofer-Wohl (2017) use the SIPP,
for example.

72

SUNHAM KIM

Figure B.1: Optimal Education Choice by utility form(a) Logarithmic

Child Ability 'Patental e, given parental

Patental , given parental e(b) Linear

Child Ability 'Patental e, given parental

Patental , given parental e(c) CRRA, ρ = 1.1

Child Ability 'Patental e, given parental

Patental , given parental e(d) CRRA, ρ = 0.9

Child Ability 'Patental e, given parental

Patental , given parental e(e) QuadraticChild Ability 'Patental e, given parental

Patental , given parental e73

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

year flows has to combine two data sources. The other approach with the 1-year
flows incurs a “simulation” to translate the observed 1-year flows into 5-year ones.
I start from the 5-year flow approach. The ideal dataset is at least two consecutive series of the 5-year migration flows from a single source. To my knowledge,
there is no single dataset that comes with consecutive 5-year migration flows. Two
census-type datasets have the 5-year retrospective question (“State of residence 5
years ago”): The decennial census (1980, 1990, 2000) and the March CPS-ASEC(Current Population Survey-Annual Social and Economic Supplement) (1985,
1995, 2005, 2015). For the 1-year question, the literature mostly uses the American Community Survey (2001-2020) and March CPS (1985-2020). (Table C.1) The
Table C.1: Migration Data Sources
Data Sources by Year
A. 5-year
Reference Year
Retrospective
Decennial Census
ASEC (March CPS)
B. 1-year
ACS
CPS, ASEC

1980 1985
1975 1980
O
O

1990
1985
O

1995
1990

2000
1995
O

O

2005
2000
O

2010
2005

2015
2010
O

2001 - 2020
1985 - 20201990 and 2000 census are about migration between 1985-1990 and 1995-2000,
respectively. The series lacks the information for 1990-1995. Similarly, the ASEC
has 1980-85, 1990-95, 2000-05, and 2010-15. Hence, no single data source is able
to construct a consecutive series alone. Alternating the Census and ASEC the
only way to have a complete series of observed 5-year flows. In sum, the 5-year
migration flows are available, but not from a single source. I name this combined
sample the “alternating sample.”

74

SUNHAM KIM

Eckert and Kleineberg (2021) use the 1-year flows from the 2006-2010 ACS
data in order to construct the 5-year flows. (Appendix B.1) Specifically, they takethe following accounting identity seriously:
′

′

k
k
Nt+1
= Mkk
t−1,t Nt(42)′kwhere Ntk is the population in origin k before moving and Nt+1
is the population′in destination k ′ after moving. Mkk
t−1,t denotes the the 1-year moving matrix, i.e.,
the population share that moves from origin m to destination m′ between yearst − 1 to t. These objects are observed in the data. They simulate the 1-year moving
matrix forward five times to construct 5-year moving flows as:′′k
Ntk = Mkk
t−5,t Nt−5 ≈

5
hY

′
Mkk
t,t−d

i
k
Nt−5

(43)

d=1

I follow their practice to construct the 5-year flows using the CPS-ASEC. As the
1-year migration information is available for more than thirty years and allows
me flexibly choosing the initial year of forward iteration. Two samples can beconstructed from the 1-year approach. First, the simulated 5-year flows can beused to fill the gaps in the ASEC 5-year information. It constitutes a series of
5-year flows from 1975-2010, where the simulated flows are used for 1985-90,
1995-2000, and 2005-10. Second, a series can be fully simulated from 1985-2010.
I call each sample the “simulated 5-year sample.” and the “simulated all-year
sample,” respectively.
All in all, there is no one perfect dataset for migration flows. Table C.2 compares the three constructed samples. Fortunately, the simulated flows are able
to predict the observed flows with R2 > 0.99 for any 5-year flows. (Table C.3)
The regression coefficients imply the simulated flows are 2-3% smaller than theobserved flows while the constant is effectively zero and precisely estimated. I

75

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

use the simulated all-year sample as a main dataset. The aggregate headcounts
are normalized to 100 for each year in order to control for population growth.
Table C.2: Three Samples of 5-year Migration Flows
Simulated All-year Sample

Simulated 5-year Sample

Alternating Sample

CPS-ASEC
Yes, fully
Annual1985-2010 (26 periods)

CPS-ASEC
Yes, partially
Demi-decennial
1985-2010 (6 periods)

CPS-ASEC and Census
No
Demi-decennial
1985-2000 ( 4 periods)

SourceSimulation
Raw Data Frequency
Availablity

Table C.3: Observed and Simulated 5-year Migration Flows(1)
All ASEC(2)All Census(3)85-90, Census(4)90-95, ASEC

(5)95-00, Census(6)00-05, ASEC

(7)
10-15, ASEC

Simulated Flows

1.017***(0.008)1.011***(0.006)

1.026***(0.012)

1.022***(0.018)0.998***(0.014)1.016***(0.018)1.015***(0.004)

Constant-0.108***(0.019)-0.070(0.067)-0.160***(0.033)-0.136***(0.045)0.013(0.029)-0.097***(0.031)-0.093***(0.024)R-squared
Obs.

0.999
48

0.999
32

0.999
16

0.998
16

0.999
16

0.999
16

1.000
16

Standard errors in parentheses. ∗p < 0.1, ∗ ∗ p < 0.05, ∗ ∗ ∗p < 0.01

76

D.

SUNHAM KIM

Estimation and Computation

D.1 Moving Costs and the Migration Elasticity: Two-Stage PPML
Estimation
I estimate the migration elasticity 1/ν and the moving cost parameter τ k,l using thetwo-stage Poisson Pseudo-Maximum Likelihood (PPML) estimation.21 It derives
the regression equations from the household’s problem. I derive the estimating
equations using Equations (4) and (5), abstracting away from the special periods
such as parental, pre-parental, and terminal.
In the first step, I recast (5) as
Ntq (k; s)mqt (k, l; s) = exp

hβ
ν

τ
β q+1
1
q+1
Vt+1
(l; s) − Vt+1
(k; s) + log Ntq (k; s) − Ωqt (k, s) −
ν

ν

k,l i

ν
(44)

where Ntq (k; s)mqt (k, l; s) is the headcount of individuals moving from k to l between t and t + 1, for each q and s. The option value Ωqt (k, s) is defined as:Ωqt (k, s) ≡

X

h
i ν1
q+1
q+1
exp βVt+1(l; s) − βVt+1(k; s) − τ k,ll∈K

Equation (44) can be rearranged as a PPML regression as follows, which is Equation (37):hτ k,l iZtq (k, l; s) = exp destqt (l; s) + origtq (k; s) −+ ζtq (k, l; s)
ν
21

(45)

The PPML approach is first employed for estimating the gravity equation in trade literature.
Artuc (2013) combined it with dynamic discrete choice framework in the context of sectoral
switching. Artuç and McLaren (2015) applied the framework to sectoral and occupational switching in the United States. Recent studies applied it to geographical switching. (e.g. Caliendo et al.
(2021), Suzuki (2021) and Eckert and Kleineberg (2021)).

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

77

where Ztq (k, l; s) is the headcount N m, destqt (l; s) is a destination fixed effect,
origtq (k; s) is an origin fixed effect, τ k,l /ν is the moving cost normalized by the
migration elasticity, and ζtq (k, l; s) is an error term. As the fixed effects are identified up to a constant of normalization, I set destqt (1; s) = 0, or set region 1 as thebase category for the destination fixed effect. Specifically, the fixed effects aredefined as follows:
β
β
q+1
q+1
Et Vt+1(l; s) − Et Vt+1
(1; s)
ν
ν

β
1
q+1
q+1
origtq (k; s) ≡ − Et Vt+1
(k; s) − Et Vt+1
(1; s) + log Ntq (k; s) − Ωqt (k, s)ννdestqt (l; s) ≡The normalization with respect to τ is implied by the assumed cost structure
τ k,k = 0∀k. Estimating (45) using a PPML is the first step of the procedure.
The second step uses the Bellman equation. Equation (4) can be rearrangedas:q+1
Et Vt+1(k, s) = u

 I k (s) q+2
t+1+ βEt Vt+2
(k, s) + Ωq+1
t+1 (k, s)
k
Pt+1

(46)

Consolidating the definitions of Ω, dest, orig and (46), I obtain the following estimation equation:
κqt (k, s) = Dtq (s) +k(s) β  It+1+ ζtq (k, s)
u
k
ν
Pt+1

(47)

where the dependent variable κqt (k, s) is a function of the fixed effect estimates
Ik

and Ntq (k, s), Dtq (k, s) is time dummy and ζtq (k, s) is the error term. As t+1
Pk

(s)

is

t+1

data, I can estimate (47) with the assumption on the form of u(·). The real income
is instrumented by the lagged values as in ACM, Artuç and McLaren (2015), andCDP. The literature mostly uses the two-period lagged values, but I use the oneperiod one, which is effectively five years due to the time specification of the

78

SUNHAM KIM

model. The dependent variable and time dummy expressions are as follows.
q+1
κqt (k, s) ≡ destqt (k, s) + βorigtq (k, s) − β log Nt+1
(k, s)
β
q+2
q+1
Dtq (s) ≡Et Vt+2(1, s) − Et Vt+1
(1, s)
ν

There are two issues with applying the PPML estimation procedure for mymodel. First, in previous studies, dynamic discrete choice framework is translated into the structural regression equations based on two premises: (i) workers
live forever; (ii) all workers solve the same problem for every period, facing no
intra-period tradeoff.22 In my model, the agents are finite-lived and the worker’s
problem changes across periods. The general idea of the PPML procedure isvalid while the estimating equations vary by age. Parents face the consumptioneducational spending tradeoff, pre-parents take expectation with respect to both
idiosyncratic taste shock u and kid’s ability draw ε before making relocation decision, and retiring workers hold no option value.
For now, I abstract away from the lifecycle structure and follow the estimation
procedure as if the model agents lives forever and solve the same problem forevery period. I pool the sample across ages and all state variables other thaneducation. That is, I use gross migration flows and real incomes by region andeducation for the first and second stages, respectively. (See Appendix C.2 for thethree sample description.)
Table D.1 report the summary statistics of τok,l /ν estimated in the first stage.
The distribution is comparable across samples. Every correlation coefficient
between the three sets of estimates is higher than 0.95.
22

CDP extends their framework to incorporate an elastic labor supply. All workers face the samelabor-leisure choice for every period where the optimal leisure choice is up to the time-constantutility parameter with respect to leisure.

79

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

Table D.1: The Moving Costs

Mean
St.DevMin
Max

Simulated All-year Sample
≤ HS
≥ SC

Simulated 5-year Sample
≤ HS
≥ SC

Alternating Sample
≤ HS
≥ SC

7.539
1.473
3.745
12.017

7.236
1.740
3.083
11.911

7.234
1.744
3.081
12.139

6.551
1.174
3.580
10.848

6.440
1.287
3.323
11.003

6.450
1.304
3.307
10.712See Appendix C.2 for the sample definitions. HS and SC stand for “High School or less”
and “Some College or More”, respectively.

Table D.2 reports the second stage estimates. The all-year sample estimates
are moderately higher than the simulated and alternating 5-year estimates andsignificant at 1%. The two 5-year estimates are all significant but the sizes aredifferent. I take the simulated sample estimate for all workers (Column 1), whichis from the largest sample. The difference between HS or Less (Column 2) andSome College or More (Column 3) estimates is statistically insignificant. The
Alternating sample estimates are insignificant except the HS or Less sample(Column 7). It could be a result of the small sample size: the identification is from
3 periods, which is likely to be too short, especially compared to the 20 periods ofthe all-year simulated sample.
Table D.2: The Migration Elasticity
Simulated Sample, All Years

Simulated Sample, 5 Years

Alternating Sample

(1)
All

(2)
HS or Less

(3)
SC or More

(4)
All

(5)
HS or Less

(6)
SC or More

(7)
All

(8)
HS or Less

(9)
SC or More

1/ν

0.617***(0.142)0.546***(0.188)0.643***(0.141)0.385***(0.137)0.417**(0.171)0.557***(0.161)0.246(0.153)0.384**(0.185)0.243(0.183)

N

1880

940

940

470

235

235

282

141

141

See Appendix C.2 for the sample definitions. ∗p < 0.1, ∗ ∗ p < 0.05, ∗ ∗ ∗p < 0.01

80

D.2

SUNHAM KIM

Calibration Algorithm

1. Set the convergence tolerance.
2. Take the target moments y k , wok , N k , national college share, and national
income dispersion from the data. Take the external and PPML-estimatedparameters.
3. Guess hk , Θk , Akc , ak . Akn is determined by Akc + Akn = 1.
k
k
k
k
k
4. Back out LkD
o from y , wo , Θ , and Ao . By doing so, I automatically target y

and wk .
5. Solve the model, calculate the model-predicted moments, and regional
labor supply LkSo .
6. Check the distance between data and model predictions for the remaining
moments. Also check if all regional markets are cleared.
7. Iterate until converge. I use lsqnonlin solver of MATLAB.

D.3

Solution Method of the Household’s Problem

This subsection describes a computation strategy for the household’s problem. I
propose a solution and the corresponding algorithm. I show that the proposedsolution delivers a dramatic speed gain and is as accurate as the conventional
value function iteration.
D.3.1

Challenge

I start with the computational challenges. The dynamic spatial general equilibrium framework is a high-dimensional problem by nature. The number of states

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

81

grows with the number of worker types and regions. Even if I strictly limit the
number of worker types (inner loop), the model eventually has to deal with theat least 502 combinations of 50 U.S. states by each type, to account for migration
choice. Since I cannot reduce the number of states, I have to accelerate the inner
loop calculation.
The model’s inner loop is non-standard. It is a finite-horizon problem, which
is typically solved quickly by backward induction demanding no iteration at all.
However, the model’s parents have to take care of their children in the middle oftheir life. The initial-period value V 0 thus appears in the parents’ Bellman equation. The structure demands an iterative approach so does computation time.
A speed of the conventional value function iteration (VFI) effectively prevents
the general equilibrium analysis. Parallelization was insufficient to overcome thedesign limit.
D.3.2

Solution Background: a Brief “Anatomy” of the Model Structure

The idea exploits the finite-horizon OLG structure. To set the stage, I consider
the following finite-horizon dynamic programming problem. It is the basic OLG
model with no geography and occupation. The age index q is written in general,so qR is the retirement age, qP is the parental age. All notations are the same as
before.

82

SUNHAM KIM

V qR (e, ε) = u(I)V qP (e, ε, ε′ ) = maxu(I − e′ ) + αV 0 (e′ , ε′ ) + βV qP +1 (e, ε)′e

c + e′ = w(heη ε)subject to(48)
V qP −1 (e, ε) = u(I) + Eε′ βV qP (e, ε; ε′ )V q (e, ε) = u(I) + βV q+1 (e, ε)forq∈/ {0, qP − 1, qP , qR }V 0 (e, ε) = βV 1 (e, ε)
It is useful to have the first order condition and the envelope condition from
Equation (48).
uc (I − e′ ) = αVe0′ (e′ , ε′ )[FOC] (49)

VeqP (e, ε, ε′ ) = uc (I − e′ )∆ + βVeqP +1 (e, ε)[EC] (50)where ∆ ≡ ∂I
= ηwheη−1 ε. It is worth mentioning that V q is accurate and does∂e
not demand any iteration for q > qP , given the model structure. Moreover, thefinite-horizon structure lets me backwardly construct a sequence of Veq from theterminal-period marginal value VeqR , given any policy function e′ :VeqR (e, ε) = uc (I)∆VeqP (e, ε, ε′ ) = uc (I − e′ )∆ + βVeqP +1 (e, ε)(51)

VeqP −1 (e, ε) = uc (I)∆ + Eε′ βVeqP (e, ε; ε′ )Veq (e, ε) = uc (I)∆ + βVeq+1 (e, ε)Ve0 (e, ε) = βVe1 (e, ε)for q ∈
/ {0, qP − 1, qP , qR }(52)

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

83

Recognize that (51) is the envelope condition (50). That is, the EC holds
automatically by construction. It is notable that the EC holds for any choice of e′ ,implying that the FOC and EC are isolated from each other.23 This result is not
surprising and stems from the model structure. Fundamentally, the education
investment decision is an endogenous process of worker type that would be fixedover the rest of life, making e operates as an exogenous state variable. The policy
function e′ is for their children and thus has nothing to do with the EC.
Now the question narrows down to solving the FOC. The LHS uc (I − e′ ) isreadily and inexpensively calculable. The RHS αVe0′ (e′ , ε′ ) however involves aderivative of an unknown function V 0 . It is seemingly not directly calculable andpushes me to use the conventional value function iteration. However, it can beobtained from Equation (52), for any e′ , exploiting the model structure. I simply
interpolate the backwardly constructed Ve0 with the given e′ to evaluate Ve0′ (e′ , ε′ ).
Neither backward induction nor interpolation is computationally burdensome.24
Thus, I need one and only one numerical root-finding step: solving the FOC.
D.3.3

Algorithm

The following algorithm constructs the first derivative of the value function by
backward induction, which automatically satisfies the envelope condition. It
finds the optimal investment from the FOC. It again constructs value function
backwardly. Specifically, the following iteration procedure is my solution algorithm, followed by the conventional value function iteration:
Algorithm 1: Educated Value Function Iteration (name tentative)(i) Guess e′ .
2324This is not true in most classes of dynamic programming problems, of course.
This depends on the interpolation scheme. I briefly discuss this in Appendix D.3.4

84

SUNHAM KIM(ii) Find Ve0 (e, ε) by backward induction, given e′ .
(iii) Check if e′ satisfies uc (I − e′ ) = αVe0′ (e′ , ε′ ).
Iterate on (i)-(iii) until e′ solves the FOC. Given the solved policy function, backwardly construct V .
Algorithm 2: Conventional VFI(i) Guess V 0 .
(ii) Solve for e′ that maximizes the parent’s Bellman equation (48).
(iii) Construct V̂ with the solved e′ by backward induction and update V 0 = V̂ 0 .
Iterate on (i)-(iii) until convergence V̂ = V .
While both algorithms solve the same number of maximization problems,
Algorithm 1 exploits more information from the model. It leverages the fact that
the EC holds regardless of e′ . As a result, it does not need to repeatedly evaluate
V and (i)-(iii) can be jointly implemented as a FOC rootfinding. It computes V
only once at the very end, when both Ve and e′ are obtained. On the contrary,
Algorithm 2 evaluates V in each iteration and converges to the true solution by
gradually updating V 0 .
Both require some interpolation of Ve0 (Algorithm 1) or V 0 (Algorithm 2) withrespect to e. The interpolation scheme is described in the following subsection.
D.3.4

Interpolation

I employ the shape-preserving rational function spline Hermite interpolation (CaiJudd 2012). I do not describe the detail. It is inexpensive and preserves a shape ofconcave, monotonically increasing function. The approximated function is C 1

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

85

globally, and C ∞ on each interval. It is applicable only if value function is known
to be concave and monotonically increasing. My framework meets the restriction.
Popular options such as linear and Chebyshev are not suitable for my goal. I needa curvature of Ve , so linear is not a way to go. Chebyshev is not shape-preserving.
it is successful in a number of applications in spite of the drawback. But mine did
not work with it.
D.3.5

Numerical Example

I solve a one-region version of the model using Algorithms 1 and 2. The results arecomparable, and Algorithm 1 is much faster than Algorithm 2. Both algorithms
converge to the (numerically) same fixed point and Algorithm 1 can be usedwithout concern.
Setup• The terminal age qR = 4.
• CRRA utility with risk aversion ρ = 0.9.
• Returns to educational investment η = 0.1• Stochastic discount factor β = 0.9; Altruism parameter α = 0.5• Regional HC production efficiency h = 1; Exogenous wage w = 1• Discretized grid of e: [0.06, 0.4]. unevenly distributed 50 points; clustered on
the left of the interval• Ability ε follows lognormal distribution. 16 points.
The state space for any q ̸= qP is 50 × 16 = 800. For qP , it is 50 × 162 = 12, 800. Thus,
I end up with 800 × 3 + 12, 800 = 15, 200 states. Convergence threshold is 10−10 for
both.

86
Results

SUNHAM KIM

Table D.3 reports the speed of each algorithm. Algorithm 1 takes lessthan a second, which is almost 300 times faster than Algorithm 2 with an educatedguess, which is V 0 constructed from e′0 . The slowest version is Algorithm 2 from ablind guess, V 0 = 0. Algorithm 1 is as 400 times faster than it. Since Algorithm 2
converges to the same fixed point by the contraction mapping theorem, I use theresults with the educated guess for the following comparison.
Figures D.1 and D.2 compare V 0 and e′ from each algorithm. In each subplot,a higher curve is from a higher-ability parent. For e′ , which has an additional
dimension ε′ , I fixed it across all curves. The differences between the algorithms
are tiny.
In sum, Algorithm 1 is fast and reliable.
Table D.3: Speed of each algorithm

CPU Time (Absolute)
CPU Time (Relative)

Algo 1

Algo 2 (blind guess)

0.7
1

277.4
396.3

Figure D.1: Model Solution: Value Function
Algo 2

Algo 1
44

44

43

43

42

42

0

0.2

Patental e, given parental

0.4

0

1

10-3 Algo 2 - Algo 1

0

0.2

Patental e

0.4

-1
0

0.2

Patental e

0.4

87

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

Figure D.2: Model Solution: Policy Function
Algo 2

Algo 1
0.3

0.3

0.2

0.2

0.1
0

0.2

Patental e, given parental

D.4

0.4

0.1
0

1

10-3 Algo 2 - Algo 1

0

0.2

Patental e

0.4

-1
0

0.2

0.4

Patental e

Extension to the Spatial Economy

Algorithm 1 completely separates the value function and its derivative. The
separation does not hold once the migration is allowed, because the derivative
now involves the levels of the next period’s value function due to the migration
propensity. I propose a workaround based on fixed-point iteration. Although thealgorithm does not work as is, a simple modification revives it.
To state the problem, I write the Ve sequence in spatial economy:
VeqR (k; s) = uc (I)∆
VeqP (k, ε′ ; s) = uc (I − e′ )∆ + E βVeqP +1 (k ′ , s)
VeqP −1 (k, s) = uc (I)∆ + E Eε′ βVeqP (k ′ , ε′ ; s)Veq (k, s) = uc (I)∆ + E βVeq+1 (k ′ ; s)for q ∈
/ {0, qP − 1, qP , qR }
Ve0 (k, s) = E βVe1 (k ′ ; s)where s ≡ (kB , e, ε) is the type-defining state vector, the expectation is taken Pwith respect to migration, i.e., E βVeq+1 (k ′ , s) = k′ mq (k, k ′ ; s)βVeq+1 (k ′ ; s), andmq (k, k ′ ; s) is the migration propensity. It is clear that mq is a function of V q+1 (the88

SUNHAM KIM

formula is omitted), which implies we need V qP , thus V 0 , to obtain Veq for any
q < qP . Therefore, the proposed strategy does not work as is.
It is useful to clarify two key properties of the finite-horizon problem. First, we
can construct V from any guess of e′ as in conventional policy iteration, without
knowing Ve . Second, the Ve sequence can be calculated for any mq , so is for any
V . Such evaluation of Ve is possible because there is no feedback from Ve to V
once e′ is given. It is another implication of the separability that my modification
leverages. If V is correct, the Ve sequence is correct as well, and we can move onto the FOC evaluation.25
In short, I guess e′ then construct the corresponding V prior to Ve . We canuse the constructed V to obtain m and Ve . The FOC can be evaluated using
those. The last building block is to calculate V consistent with the guess of e′ . It
is manageable via fixed-point iteration. Given any policy e′ , the finite-horizon
dynamic programming problem can be reformulated to an equivalent fixed-point
problem:
V0 =g V0and then to use the iteration: choose an initial guess V 0(0) , compute a sequence
V 0(n+1) = g V 0(n)and obtain V 0(n) → V 0 . Here, g(·) is defined by a discounted sum of future values,
thus increasing, concave, and differentiable. It is straightforward to show that
there is a unique solution V 0 for any given e′ . Moreover, this new step incurs
little additional computational burden as it involves neither derivatives Veq nor
numerical root-finding.
25

Feeding an incorrect V is not a concern. If the given V were incorrect, the generated sequence
would be simply incorrect. The error stems from using mq inconsistent with the true valuefunction. The solver would just move on to the next guess of e′ in such a case.

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY89

In sum, the FP iteration delivers the guess-consistent V . Armed with that, we
can find Ve and evaluate the FOC. The following is the revised algorithm.
Algorithm 3(i) Guess e′ .
(ii) Find V by fixed-point iteration.
(iii) Find Ve0 (e, ε) by backward induction, with V and e′ .
(iv) Check if e′ satisfies uc (I − e′ ) = αVe0′ (e′ , ε′ ).
Iterate on (i)-(iv) until e′ solves the FOC. Both V and e′ are obtained once the FOC
is solved.
As in the simple case of no migration, this algorithm converges to the conventional VFI solution. The speed gain is again significant. I found Algorithm 3 is
255x faster than the VFI in an illustrative two-region case. Results are available
upon request.

90

SUNHAM KIM

Figure E.1: ANI by Education: Levels(a) High School or Less(b) Some College or MoreE.

Additional Quantification Results

E.1

TablesE.2

FiguresHUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

91

Table E.1: Calibration Fit: U.S. States
AL
AZ
AR
CA
CO
CT
FL
GA
HI
ID
IL
IN
IA
KS
KY
LA
ME
MD
MA
MI
MN
MS
MO
MT
NE
NV
NH
NJ
NM
NY
NC
ND
OH
OK
OR
PA
RI
SC
SD
TN
TX
UT
VT
VA
WA
WV
WI

Per Capita GDP (Model)
1.413
1.727
1.247
1.961
1.538
2.286
1.831
1.679
1.841
1.258
1.935
1.592
1.504
1.318
1.432
1.651
1.377
1.976
1.968
1.631
1.603
1.225
1.489
1.03
1.566
1.977
1.565
2.116
1.199
2.254
1.713
1.04
1.623
1.15
1.514
1.659
1.778
1.483
1.435
1.484
1.699
1.445
1.204
1.969
1.998
1.277
1.456

Data
1.413
1.727
1.247
1.961
1.538
2.286
1.831
1.679
1.841
1.258
1.935
1.592
1.504
1.318
1.432
1.651
1.377
1.976
1.968
1.631
1.603
1.225
1.489
1.03
1.566
1.977
1.565
2.116
1.199
2.254
1.713
1.04
1.623
1.15
1.514
1.659
1.778
1.483
1.435
1.484
1.699
1.445
1.204
1.969
1.998
1.277
1.456

All state-level moments are matched exactly.

Avg wage (Model)
0.714 1.081
0.639 1.231
0.654 1.051
0.73 1.349
0.775 1.26
0.818 1.394
0.727 1.169
0.747 1.171
0.775 1.073
0.729 1.047
0.756 1.21
0.76 1.144
0.743 1.036
0.718 1.098
0.744 1.055
0.775 1.139
0.704 1.078
0.837 1.388
0.815 1.368
0.721 1.174
0.751 1.178
0.674 0.968
0.716 1.113
0.721 0.848
0.762 1.055
0.815 1.19
0.853 1.217
0.848 1.495
0.675 1.075
0.751 1.254
0.668 1.09
0.693 0.947
0.74 1.103
0.71 1.028
0.728 1.138
0.758 1.183
0.763 1.193
0.693 1.034
0.705 0.972
0.67 1.046
0.704 1.169
0.74 1.145
0.782 1.036
0.797 1.331
0.84 1.213
0.702 1.075
0.798 1.112(Data)
0.714 1.081
0.639 1.231
0.654 1.051
0.73 1.349
0.775 1.26
0.818 1.394
0.727 1.169
0.747 1.171
0.775 1.073
0.729 1.047
0.756 1.21
0.76 1.144
0.743 1.036
0.718 1.098
0.744 1.055
0.775 1.139
0.704 1.078
0.837 1.388
0.815 1.368
0.721 1.174
0.751 1.178
0.674 0.968
0.716 1.113
0.721 0.848
0.762 1.055
0.815 1.19
0.853 1.217
0.848 1.495
0.675 1.075
0.751 1.254
0.668 1.09
0.693 0.947
0.74 1.103
0.71 1.028
0.728 1.138
0.758 1.183
0.763 1.193
0.693 1.034
0.705 0.972
0.67 1.046
0.704 1.169
0.74 1.145
0.782 1.036
0.797 1.331
0.84 1.213
0.702 1.075
0.798 1.112

Pop Shares (Model)
0.017
0.017
0.01
0.088
0.017
0.013
0.049
0.032
0.004
0.005
0.045
0.026
0.012
0.011
0.016
0.016
0.005
0.019
0.024
0.04
0.022
0.01
0.023
0.003
0.007
0.007
0.006
0.028
0.006
0.057
0.033
0.003
0.048
0.013
0.013
0.049
0.004
0.016
0.003
0.023
0.069
0.007
0.003
0.028
0.022
0.007
0.024

(Data)
0.017
0.017
0.01
0.088
0.017
0.013
0.049
0.032
0.004
0.005
0.045
0.026
0.012
0.011
0.016
0.016
0.005
0.019
0.024
0.04
0.022
0.01
0.023
0.003
0.007
0.007
0.006
0.028
0.006
0.057
0.033
0.003
0.048
0.013
0.013
0.049
0.004
0.016
0.003
0.023
0.069
0.007
0.003
0.028
0.022
0.007
0.024

92

SUNHAM KIM

Table E.2: Calibrated Regional Fundamentals
State

Skill Productivity h

TFP Θ

AL
AZ
AR
CA
CO
CT
FL
GA
HI
ID
IL
IN
IA
KS
KY
LA
ME
MD
MA
MI
MN
MS
MO
MT
NE
NV
NH
NJ
NM
NY
NC
ND
OH
OK
OR
PA
RI
SC
SD
TN
TX
UT
VT
VA

1.76001.8832
1.67102.0534
1.84672.18712.01781.90642.01871.74092.04301.8631
1.88141.71491.80201.91291.76942.02452.0581
1.85701.87091.66981.80601.6283
1.93272.09461.90992.06001.59332.1605
1.88851.58571.86941.5903
1.83281.8652
1.94491.77251.85591.7703
1.88601.7862
1.68842.01281.67111.70661.5332
1.8074
1.68862.0763
1.89291.81671.88831.51461.97271.81401.77651.54251.7305
1.8486
1.58851.92321.91141.8026
1.79851.5466
1.71161.38051.81962.09201.7324
1.98361.40992.09831.81551.35751.87161.50311.7265
1.83401.84331.72091.73681.71351.82151.6713
1.4695
1.9260Amenity a CES shifter Ac
0.69630.7942
0.69890.79050.72750.76350.72140.7171
0.71150.66330.72290.68780.63990.71450.65290.68650.70380.75960.75040.72190.70590.66810.70600.5497
0.63630.67980.68000.77420.73860.74980.73870.64450.67550.65960.68930.71040.7262
0.69840.62710.71360.73720.70020.64680.76040.69630.7942
0.69890.79050.72750.76350.72140.7171
0.71150.66330.72290.68780.63990.71450.65290.68650.70380.75960.75040.72190.70590.66810.70600.5497
0.63630.67980.68000.77420.73860.74980.73870.64450.67550.65960.68930.71040.7262
0.69840.62710.71360.73720.70020.64680.7604

93

HUMAN CAPITAL PRODUCTION IN A SPATIAL ECONOMY

Table E.3: Model Validation: TFP and Amenities
TFP
Herkenhoff et al. (2018)
1980
1990
2000
2014

0.280
(0.060)0.703(0.000)0.743(0.000)0.580(0.000)Amenities
Herkenhoff et al. (2018) Amenities, Others-0.225(0.132)-0.076(0.615)-0.321(0.030)-0.550(0.000)0.500(0.000)0.415(0.004)0.176(0.235)Spearman rank correlation coefficients. P-values in parentheses. For “Others”,
the 1980 and 1990 estimates are taken from Gabriel et al. (2003). The 2000 estimate is taken from Albouy (2008).

Figure E.2: RHCPC by Education: Levels(a) High School or Less(b) Some College or More